Wikiversity:Request custodian action

<mediawiki xmlns="http://www.mediawiki.org/xml/export-0.11/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.mediawiki.org/xml/export-0.11/ http://www.mediawiki.org/xml/export-0.11.xsd" version="0.11" xml:lang="en">
  <siteinfo>
    <sitename>Wikiversity</sitename>
    <dbname>enwikiversity</dbname>
    <base>https://en.wikiversity.org/wiki/Wikiversity:Main_Page</base>
    <generator>MediaWiki 1.47.0-wmf.21</generator>
    <case>first-letter</case>
    <namespaces>
      <namespace key="-2" case="first-letter">Media</namespace>
      <namespace key="-1" case="first-letter">Special</namespace>
      <namespace key="0" case="first-letter" />
      <namespace key="1" case="first-letter">Talk</namespace>
      <namespace key="2" case="first-letter">User</namespace>
      <namespace key="3" case="first-letter">User talk</namespace>
      <namespace key="4" case="first-letter">Wikiversity</namespace>
      <namespace key="5" case="first-letter">Wikiversity talk</namespace>
      <namespace key="6" case="first-letter">File</namespace>
      <namespace key="7" case="first-letter">File talk</namespace>
      <namespace key="8" case="first-letter">MediaWiki</namespace>
      <namespace key="9" case="first-letter">MediaWiki talk</namespace>
      <namespace key="10" case="first-letter">Template</namespace>
      <namespace key="11" case="first-letter">Template talk</namespace>
      <namespace key="12" case="first-letter">Help</namespace>
      <namespace key="13" case="first-letter">Help talk</namespace>
      <namespace key="14" case="first-letter">Category</namespace>
      <namespace key="15" case="first-letter">Category talk</namespace>
      <namespace key="100" case="first-letter">School</namespace>
      <namespace key="101" case="first-letter">School talk</namespace>
      <namespace key="102" case="first-letter">Portal</namespace>
      <namespace key="103" case="first-letter">Portal talk</namespace>
      <namespace key="104" case="first-letter">Topic</namespace>
      <namespace key="105" case="first-letter">Topic talk</namespace>
      <namespace key="106" case="first-letter">Collection</namespace>
      <namespace key="107" case="first-letter">Collection talk</namespace>
      <namespace key="118" case="first-letter">Draft</namespace>
      <namespace key="119" case="first-letter">Draft talk</namespace>
      <namespace key="710" case="first-letter">TimedText</namespace>
      <namespace key="711" case="first-letter">TimedText talk</namespace>
      <namespace key="828" case="first-letter">Module</namespace>
      <namespace key="829" case="first-letter">Module talk</namespace>
      <namespace key="1728" case="first-letter">Event</namespace>
      <namespace key="1729" case="first-letter">Event talk</namespace>
    </namespaces>
  </siteinfo>
  <page>
    <title>Wikiversity:Request custodian action</title>
    <ns>4</ns>
    <id>75745</id>
    <revision>
      <id>2834674</id>
      <parentid>2834159</parentid>
      <timestamp>2026-09-27T12:57:15Z</timestamp>
      <contributor>
        <username>Petrosa51</username>
        <id>3111696</id>
      </contributor>
      <comment>/* I am trying to publish a biography */ new section</comment>
      <origin>2834674</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8027" sha1="tjmusc5s8yqu9jlkvxvyuyw2d4er96v" xml:space="preserve">{{/Header}}

==Review changes to [[Special:AbuseFilter/4]]==
{{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:44, 25 May 2026 (UTC)
: I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC)

== Please unban ==

I want to create a new page (a pro/contra discussion page [[Are humans omnivores or herbivores?|like this one]]); initially it included an entry in the &quot;== External Links ==&quot; section, so it got blocked. I tried using a doi.org link; unfortunately that got blocked, too. Then I tried without, and it said I tried too often…
Please unban my user so I can create the page (without an external link…). In general, please consider adding doi.org to a whitelist. DOI links are virtually all academic, so how bad can that possibly be?
Thank you! :-) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:02, 15 August 2026 (UTC)
: Thanks for letting us know. Some suggestion:
:* Although you can edit anonymously, it is better to create an account and log in - that will give you more permissions
:* You are hitting some spam filters, but the system will learn to trust you if you login and engage in some constructive editing (e.g., edit and improve a page before before adding external links); basically, it doesn't trust you yet. We don't have a lot of volunteer admins so it isn't perfect (there are some false positives), but it prevents a lot of clean-up.
: 
: But now for the not-so-good news. The Wikiversity community recently decide to stop Wikidebates because there were some problematic topics and responses. However, if it is a good topic, there's likely some way to do something similar that would be accepted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 15 August 2026 (UTC)
::My debate question is: &quot;Are humans monogamous or polygamous?&quot; (In my opinion, that makes it very analogous to the diet question I liked to.) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:37, 15 August 2026 (UTC)
::: This question is an example of a potentially educationally relevant topic to have as a learning project in debate format, but currently debates are not permitted (see [[Wikidebates]] for more info). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:03, 15 August 2026 (UTC)
::::I understand. I would generally find it noteworthy that a debate club which used to pretend to be liberal would so readily admit to the opposite and shut itself down, simply because – I'm assuming – someone asked the forbidden questions of our time or questioned the truisms of our time, which stirred up some emotions in the overly sensitive. That, of course, is not what the enlightenment taught us, nor is it in line with what ''actually'' liberal academia is all about. However, it is no surprise to me at all for a socialist project of the Wikimedia realm. I'll go and find myself some better people. Thank you and have a good day, Jtneill. [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 12:10, 15 August 2026 (UTC)

== Request to review AbuseFilter block on a student research draft ==

Hello. I am a new contributor attempting to create [[Draft:Anti-IL-6 scFv–RDVLPGT fusion concept]], a clearly labeled student research project. The draft separates cited evidence from an unvalidated design hypothesis, states the central limitation (extracellular anti-IL-6 scFv versus intracellular MyD88), includes falsifiable expectations, and uses external links only for DOI references, RCSB PDB 4DOM, and an NCBI iCn3D structural-context view. The edit was disallowed by the rule “New user creating page with external link.” Could a custodian please review the draft and allow the constructive edit, or advise the appropriate way to publish it without removing its verifiable sources? Thank you. [[User:Jayhan0302|Jayhan0302]] ([[User talk:Jayhan0302|discuss]] • [[Special:Contributions/Jayhan0302|contribs]]) 08:47, 22 August 2026 (UTC)
: Hi Jayhan,
: Because you're a new user, the system wants to learn to trust you, so try editing at first without including external links. Then, after a while, it will realise that you're to be trusted, and then you'll be able to add external links.
: I hope that helps.
: Sincerely,&lt;br&gt; James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:04, 22 August 2026 (UTC)
:This is a little embarrassing, but I've never been an EditFilter whiz, so I just added the links myself. You will be able to add links yourself after you have made &lt;var&gt;x&lt;/var&gt; edits in &lt;var&gt;y&lt;/var&gt; days. I can never recall the exact numbers, but they are very small. Let me know if you need more help. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:06, 22 August 2026 (UTC)

== Report concerning [[special:contribs/Bśadów$ki112|Bśadów$ki112]] ==
'''Reasons''': Long-term abuse. Impersonation of [[:sulutil:Bsadowski1|Bsadowski1]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 19:19, 28 August 2026 (UTC)

:{{done}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 19:20, 28 August 2026 (UTC)

== Custodianship ==
proofreading and summarizing I excel in  social science and current events history political science are my areas of past study. Could I be custodian of primarily pages dealing with social science and politics? I have experience editing am a advanced reader and can accept more than one point of view on almost any topic the point is to see the ethical standard and protection of people of sacred belief of food equitable relationships and not accept any point of view that includes towards a laissez faire attitude on abuse. To me it's what we can do as individuals,as community,as friends as family not what harm is possible for that is off the boards not acceptable. I'll keep a open mind and not stifle any dissent. My interest is also in assuring the complete construction of explanatory paragraphs,proper wording and generally good punctuation.  [[Special:Contributions/~2026-50875-97|~2026-50875-97]] ([[User talk:~2026-50875-97|talk]]) 15:27, 21 September 2026 (UTC)

:What you posted here does not require custodian action and show some poor judgement: This board is for escalated issues that a standard user can't perform. What you are basically writing here is &quot;I would like to be a custodian&quot;. There is a process for that, but step one would be having an actual account to edit from, so do that first, then edit here as a non-custodian in a helpful manner, and then make the request if you'd still like that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:38, 21 September 2026 (UTC)

== I am trying to publish a biography ==
I am a beginner, I cannot succeed. [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 12:57, 27 September 2026 (UTC)</text>
      <sha1>tjmusc5s8yqu9jlkvxvyuyw2d4er96v</sha1>
    </revision>
    <revision>
      <id>2834685</id>
      <parentid>2834674</parentid>
      <timestamp>2026-09-27T16:24:32Z</timestamp>
      <contributor>
        <username>Koavf</username>
        <id>147</id>
      </contributor>
      <comment>/* I am trying to publish a biography */ Reply</comment>
      <origin>2834685</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8703" sha1="t87megj5a64oswv71quwhqqkvxnvejs" xml:space="preserve">{{/Header}}

==Review changes to [[Special:AbuseFilter/4]]==
{{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:44, 25 May 2026 (UTC)
: I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC)

== Please unban ==

I want to create a new page (a pro/contra discussion page [[Are humans omnivores or herbivores?|like this one]]); initially it included an entry in the &quot;== External Links ==&quot; section, so it got blocked. I tried using a doi.org link; unfortunately that got blocked, too. Then I tried without, and it said I tried too often…
Please unban my user so I can create the page (without an external link…). In general, please consider adding doi.org to a whitelist. DOI links are virtually all academic, so how bad can that possibly be?
Thank you! :-) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:02, 15 August 2026 (UTC)
: Thanks for letting us know. Some suggestion:
:* Although you can edit anonymously, it is better to create an account and log in - that will give you more permissions
:* You are hitting some spam filters, but the system will learn to trust you if you login and engage in some constructive editing (e.g., edit and improve a page before before adding external links); basically, it doesn't trust you yet. We don't have a lot of volunteer admins so it isn't perfect (there are some false positives), but it prevents a lot of clean-up.
: 
: But now for the not-so-good news. The Wikiversity community recently decide to stop Wikidebates because there were some problematic topics and responses. However, if it is a good topic, there's likely some way to do something similar that would be accepted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 15 August 2026 (UTC)
::My debate question is: &quot;Are humans monogamous or polygamous?&quot; (In my opinion, that makes it very analogous to the diet question I liked to.) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:37, 15 August 2026 (UTC)
::: This question is an example of a potentially educationally relevant topic to have as a learning project in debate format, but currently debates are not permitted (see [[Wikidebates]] for more info). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:03, 15 August 2026 (UTC)
::::I understand. I would generally find it noteworthy that a debate club which used to pretend to be liberal would so readily admit to the opposite and shut itself down, simply because – I'm assuming – someone asked the forbidden questions of our time or questioned the truisms of our time, which stirred up some emotions in the overly sensitive. That, of course, is not what the enlightenment taught us, nor is it in line with what ''actually'' liberal academia is all about. However, it is no surprise to me at all for a socialist project of the Wikimedia realm. I'll go and find myself some better people. Thank you and have a good day, Jtneill. [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 12:10, 15 August 2026 (UTC)

== Request to review AbuseFilter block on a student research draft ==

Hello. I am a new contributor attempting to create [[Draft:Anti-IL-6 scFv–RDVLPGT fusion concept]], a clearly labeled student research project. The draft separates cited evidence from an unvalidated design hypothesis, states the central limitation (extracellular anti-IL-6 scFv versus intracellular MyD88), includes falsifiable expectations, and uses external links only for DOI references, RCSB PDB 4DOM, and an NCBI iCn3D structural-context view. The edit was disallowed by the rule “New user creating page with external link.” Could a custodian please review the draft and allow the constructive edit, or advise the appropriate way to publish it without removing its verifiable sources? Thank you. [[User:Jayhan0302|Jayhan0302]] ([[User talk:Jayhan0302|discuss]] • [[Special:Contributions/Jayhan0302|contribs]]) 08:47, 22 August 2026 (UTC)
: Hi Jayhan,
: Because you're a new user, the system wants to learn to trust you, so try editing at first without including external links. Then, after a while, it will realise that you're to be trusted, and then you'll be able to add external links.
: I hope that helps.
: Sincerely,&lt;br&gt; James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:04, 22 August 2026 (UTC)
:This is a little embarrassing, but I've never been an EditFilter whiz, so I just added the links myself. You will be able to add links yourself after you have made &lt;var&gt;x&lt;/var&gt; edits in &lt;var&gt;y&lt;/var&gt; days. I can never recall the exact numbers, but they are very small. Let me know if you need more help. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:06, 22 August 2026 (UTC)

== Report concerning [[special:contribs/Bśadów$ki112|Bśadów$ki112]] ==
'''Reasons''': Long-term abuse. Impersonation of [[:sulutil:Bsadowski1|Bsadowski1]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 19:19, 28 August 2026 (UTC)

:{{done}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 19:20, 28 August 2026 (UTC)

== Custodianship ==
proofreading and summarizing I excel in  social science and current events history political science are my areas of past study. Could I be custodian of primarily pages dealing with social science and politics? I have experience editing am a advanced reader and can accept more than one point of view on almost any topic the point is to see the ethical standard and protection of people of sacred belief of food equitable relationships and not accept any point of view that includes towards a laissez faire attitude on abuse. To me it's what we can do as individuals,as community,as friends as family not what harm is possible for that is off the boards not acceptable. I'll keep a open mind and not stifle any dissent. My interest is also in assuring the complete construction of explanatory paragraphs,proper wording and generally good punctuation.  [[Special:Contributions/~2026-50875-97|~2026-50875-97]] ([[User talk:~2026-50875-97|talk]]) 15:27, 21 September 2026 (UTC)

:What you posted here does not require custodian action and show some poor judgement: This board is for escalated issues that a standard user can't perform. What you are basically writing here is &quot;I would like to be a custodian&quot;. There is a process for that, but step one would be having an actual account to edit from, so do that first, then edit here as a non-custodian in a helpful manner, and then make the request if you'd still like that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:38, 21 September 2026 (UTC)

== I am trying to publish a biography ==
I am a beginner, I cannot succeed. [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 12:57, 27 September 2026 (UTC)

:I posted a [[Template:welcome|welcome message]] to your talk page that gives you an overview of Wikiversity. While a biography &lt;em&gt;could&lt;/em&gt; be considered appropriate here if part of a larger exercise in learning but this is not an encyclopedia like our sibling project Wikipedia, so generally, we won't have much in the way of article-long biographies. Why were you trying to post a biography here and of whom? ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:24, 27 September 2026 (UTC)</text>
      <sha1>t87megj5a64oswv71quwhqqkvxnvejs</sha1>
    </revision>
    <revision>
      <id>2834686</id>
      <parentid>2834685</parentid>
      <timestamp>2026-09-27T16:41:34Z</timestamp>
      <contributor>
        <username>Petrosa51</username>
        <id>3111696</id>
      </contributor>
      <comment>/* I am trying to publish a biography */ Reply</comment>
      <origin>2834686</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9365" sha1="7bljvk5acnz4bchhztwnjmwp0melc7y" xml:space="preserve">{{/Header}}

==Review changes to [[Special:AbuseFilter/4]]==
{{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:44, 25 May 2026 (UTC)
: I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC)

== Please unban ==

I want to create a new page (a pro/contra discussion page [[Are humans omnivores or herbivores?|like this one]]); initially it included an entry in the &quot;== External Links ==&quot; section, so it got blocked. I tried using a doi.org link; unfortunately that got blocked, too. Then I tried without, and it said I tried too often…
Please unban my user so I can create the page (without an external link…). In general, please consider adding doi.org to a whitelist. DOI links are virtually all academic, so how bad can that possibly be?
Thank you! :-) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:02, 15 August 2026 (UTC)
: Thanks for letting us know. Some suggestion:
:* Although you can edit anonymously, it is better to create an account and log in - that will give you more permissions
:* You are hitting some spam filters, but the system will learn to trust you if you login and engage in some constructive editing (e.g., edit and improve a page before before adding external links); basically, it doesn't trust you yet. We don't have a lot of volunteer admins so it isn't perfect (there are some false positives), but it prevents a lot of clean-up.
: 
: But now for the not-so-good news. The Wikiversity community recently decide to stop Wikidebates because there were some problematic topics and responses. However, if it is a good topic, there's likely some way to do something similar that would be accepted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 15 August 2026 (UTC)
::My debate question is: &quot;Are humans monogamous or polygamous?&quot; (In my opinion, that makes it very analogous to the diet question I liked to.) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:37, 15 August 2026 (UTC)
::: This question is an example of a potentially educationally relevant topic to have as a learning project in debate format, but currently debates are not permitted (see [[Wikidebates]] for more info). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:03, 15 August 2026 (UTC)
::::I understand. I would generally find it noteworthy that a debate club which used to pretend to be liberal would so readily admit to the opposite and shut itself down, simply because – I'm assuming – someone asked the forbidden questions of our time or questioned the truisms of our time, which stirred up some emotions in the overly sensitive. That, of course, is not what the enlightenment taught us, nor is it in line with what ''actually'' liberal academia is all about. However, it is no surprise to me at all for a socialist project of the Wikimedia realm. I'll go and find myself some better people. Thank you and have a good day, Jtneill. [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 12:10, 15 August 2026 (UTC)

== Request to review AbuseFilter block on a student research draft ==

Hello. I am a new contributor attempting to create [[Draft:Anti-IL-6 scFv–RDVLPGT fusion concept]], a clearly labeled student research project. The draft separates cited evidence from an unvalidated design hypothesis, states the central limitation (extracellular anti-IL-6 scFv versus intracellular MyD88), includes falsifiable expectations, and uses external links only for DOI references, RCSB PDB 4DOM, and an NCBI iCn3D structural-context view. The edit was disallowed by the rule “New user creating page with external link.” Could a custodian please review the draft and allow the constructive edit, or advise the appropriate way to publish it without removing its verifiable sources? Thank you. [[User:Jayhan0302|Jayhan0302]] ([[User talk:Jayhan0302|discuss]] • [[Special:Contributions/Jayhan0302|contribs]]) 08:47, 22 August 2026 (UTC)
: Hi Jayhan,
: Because you're a new user, the system wants to learn to trust you, so try editing at first without including external links. Then, after a while, it will realise that you're to be trusted, and then you'll be able to add external links.
: I hope that helps.
: Sincerely,&lt;br&gt; James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:04, 22 August 2026 (UTC)
:This is a little embarrassing, but I've never been an EditFilter whiz, so I just added the links myself. You will be able to add links yourself after you have made &lt;var&gt;x&lt;/var&gt; edits in &lt;var&gt;y&lt;/var&gt; days. I can never recall the exact numbers, but they are very small. Let me know if you need more help. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:06, 22 August 2026 (UTC)

== Report concerning [[special:contribs/Bśadów$ki112|Bśadów$ki112]] ==
'''Reasons''': Long-term abuse. Impersonation of [[:sulutil:Bsadowski1|Bsadowski1]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 19:19, 28 August 2026 (UTC)

:{{done}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 19:20, 28 August 2026 (UTC)

== Custodianship ==
proofreading and summarizing I excel in  social science and current events history political science are my areas of past study. Could I be custodian of primarily pages dealing with social science and politics? I have experience editing am a advanced reader and can accept more than one point of view on almost any topic the point is to see the ethical standard and protection of people of sacred belief of food equitable relationships and not accept any point of view that includes towards a laissez faire attitude on abuse. To me it's what we can do as individuals,as community,as friends as family not what harm is possible for that is off the boards not acceptable. I'll keep a open mind and not stifle any dissent. My interest is also in assuring the complete construction of explanatory paragraphs,proper wording and generally good punctuation.  [[Special:Contributions/~2026-50875-97|~2026-50875-97]] ([[User talk:~2026-50875-97|talk]]) 15:27, 21 September 2026 (UTC)

:What you posted here does not require custodian action and show some poor judgement: This board is for escalated issues that a standard user can't perform. What you are basically writing here is &quot;I would like to be a custodian&quot;. There is a process for that, but step one would be having an actual account to edit from, so do that first, then edit here as a non-custodian in a helpful manner, and then make the request if you'd still like that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:38, 21 September 2026 (UTC)

== I am trying to publish a biography ==
I am a beginner, I cannot succeed. [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 12:57, 27 September 2026 (UTC)

:I posted a [[Template:welcome|welcome message]] to your talk page that gives you an overview of Wikiversity. While a biography &lt;em&gt;could&lt;/em&gt; be considered appropriate here if part of a larger exercise in learning but this is not an encyclopedia like our sibling project Wikipedia, so generally, we won't have much in the way of article-long biographies. Why were you trying to post a biography here and of whom? ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:24, 27 September 2026 (UTC)
::I am a writer I publish in Romania and USA, and I wanted to put on-line officially, the Bibliography, both in romanian and english, maybe I am not on the right topic pages, I did much work in Romania, and both in USA, and I am proud, but just maybe I didn't came out with the right issued page, I will do my best to make a bibliography, if I may. It was about my works, so far. I will try another day, if I succeed... Very much greetings Iulia, ro Where I can put the text, I have seen other Bibliographies, on wiki? [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 16:41, 27 September 2026 (UTC)</text>
      <sha1>7bljvk5acnz4bchhztwnjmwp0melc7y</sha1>
    </revision>
    <revision>
      <id>2834688</id>
      <parentid>2834686</parentid>
      <timestamp>2026-09-27T17:11:27Z</timestamp>
      <contributor>
        <username>Koavf</username>
        <id>147</id>
      </contributor>
      <comment>/* I am trying to publish a biography */ Reply</comment>
      <origin>2834688</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9902" sha1="ayyz2werec6jga7r8jt8eqgey8nztqa" xml:space="preserve">{{/Header}}

==Review changes to [[Special:AbuseFilter/4]]==
{{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:44, 25 May 2026 (UTC)
: I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC)

== Please unban ==

I want to create a new page (a pro/contra discussion page [[Are humans omnivores or herbivores?|like this one]]); initially it included an entry in the &quot;== External Links ==&quot; section, so it got blocked. I tried using a doi.org link; unfortunately that got blocked, too. Then I tried without, and it said I tried too often…
Please unban my user so I can create the page (without an external link…). In general, please consider adding doi.org to a whitelist. DOI links are virtually all academic, so how bad can that possibly be?
Thank you! :-) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:02, 15 August 2026 (UTC)
: Thanks for letting us know. Some suggestion:
:* Although you can edit anonymously, it is better to create an account and log in - that will give you more permissions
:* You are hitting some spam filters, but the system will learn to trust you if you login and engage in some constructive editing (e.g., edit and improve a page before before adding external links); basically, it doesn't trust you yet. We don't have a lot of volunteer admins so it isn't perfect (there are some false positives), but it prevents a lot of clean-up.
: 
: But now for the not-so-good news. The Wikiversity community recently decide to stop Wikidebates because there were some problematic topics and responses. However, if it is a good topic, there's likely some way to do something similar that would be accepted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 15 August 2026 (UTC)
::My debate question is: &quot;Are humans monogamous or polygamous?&quot; (In my opinion, that makes it very analogous to the diet question I liked to.) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:37, 15 August 2026 (UTC)
::: This question is an example of a potentially educationally relevant topic to have as a learning project in debate format, but currently debates are not permitted (see [[Wikidebates]] for more info). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:03, 15 August 2026 (UTC)
::::I understand. I would generally find it noteworthy that a debate club which used to pretend to be liberal would so readily admit to the opposite and shut itself down, simply because – I'm assuming – someone asked the forbidden questions of our time or questioned the truisms of our time, which stirred up some emotions in the overly sensitive. That, of course, is not what the enlightenment taught us, nor is it in line with what ''actually'' liberal academia is all about. However, it is no surprise to me at all for a socialist project of the Wikimedia realm. I'll go and find myself some better people. Thank you and have a good day, Jtneill. [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 12:10, 15 August 2026 (UTC)

== Request to review AbuseFilter block on a student research draft ==

Hello. I am a new contributor attempting to create [[Draft:Anti-IL-6 scFv–RDVLPGT fusion concept]], a clearly labeled student research project. The draft separates cited evidence from an unvalidated design hypothesis, states the central limitation (extracellular anti-IL-6 scFv versus intracellular MyD88), includes falsifiable expectations, and uses external links only for DOI references, RCSB PDB 4DOM, and an NCBI iCn3D structural-context view. The edit was disallowed by the rule “New user creating page with external link.” Could a custodian please review the draft and allow the constructive edit, or advise the appropriate way to publish it without removing its verifiable sources? Thank you. [[User:Jayhan0302|Jayhan0302]] ([[User talk:Jayhan0302|discuss]] • [[Special:Contributions/Jayhan0302|contribs]]) 08:47, 22 August 2026 (UTC)
: Hi Jayhan,
: Because you're a new user, the system wants to learn to trust you, so try editing at first without including external links. Then, after a while, it will realise that you're to be trusted, and then you'll be able to add external links.
: I hope that helps.
: Sincerely,&lt;br&gt; James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:04, 22 August 2026 (UTC)
:This is a little embarrassing, but I've never been an EditFilter whiz, so I just added the links myself. You will be able to add links yourself after you have made &lt;var&gt;x&lt;/var&gt; edits in &lt;var&gt;y&lt;/var&gt; days. I can never recall the exact numbers, but they are very small. Let me know if you need more help. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:06, 22 August 2026 (UTC)

== Report concerning [[special:contribs/Bśadów$ki112|Bśadów$ki112]] ==
'''Reasons''': Long-term abuse. Impersonation of [[:sulutil:Bsadowski1|Bsadowski1]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 19:19, 28 August 2026 (UTC)

:{{done}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 19:20, 28 August 2026 (UTC)

== Custodianship ==
proofreading and summarizing I excel in  social science and current events history political science are my areas of past study. Could I be custodian of primarily pages dealing with social science and politics? I have experience editing am a advanced reader and can accept more than one point of view on almost any topic the point is to see the ethical standard and protection of people of sacred belief of food equitable relationships and not accept any point of view that includes towards a laissez faire attitude on abuse. To me it's what we can do as individuals,as community,as friends as family not what harm is possible for that is off the boards not acceptable. I'll keep a open mind and not stifle any dissent. My interest is also in assuring the complete construction of explanatory paragraphs,proper wording and generally good punctuation.  [[Special:Contributions/~2026-50875-97|~2026-50875-97]] ([[User talk:~2026-50875-97|talk]]) 15:27, 21 September 2026 (UTC)

:What you posted here does not require custodian action and show some poor judgement: This board is for escalated issues that a standard user can't perform. What you are basically writing here is &quot;I would like to be a custodian&quot;. There is a process for that, but step one would be having an actual account to edit from, so do that first, then edit here as a non-custodian in a helpful manner, and then make the request if you'd still like that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:38, 21 September 2026 (UTC)

== I am trying to publish a biography ==
I am a beginner, I cannot succeed. [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 12:57, 27 September 2026 (UTC)

:I posted a [[Template:welcome|welcome message]] to your talk page that gives you an overview of Wikiversity. While a biography &lt;em&gt;could&lt;/em&gt; be considered appropriate here if part of a larger exercise in learning but this is not an encyclopedia like our sibling project Wikipedia, so generally, we won't have much in the way of article-long biographies. Why were you trying to post a biography here and of whom? ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:24, 27 September 2026 (UTC)
::I am a writer I publish in Romania and USA, and I wanted to put on-line officially, the Bibliography, both in romanian and english, maybe I am not on the right topic pages, I did much work in Romania, and both in USA, and I am proud, but just maybe I didn't came out with the right issued page, I will do my best to make a bibliography, if I may. It was about my works, so far. I will try another day, if I succeed... Very much greetings Iulia, ro Where I can put the text, I have seen other Bibliographies, on wiki? [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 16:41, 27 September 2026 (UTC)
:::If you want to contribute here and you want to mention your publications on your userpage, I think that's fine. If you just want to make a user page for vanity's sake and aren't going to actually contribute here, I don't think that's useful: we don't provide free hosting. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:11, 27 September 2026 (UTC)</text>
      <sha1>ayyz2werec6jga7r8jt8eqgey8nztqa</sha1>
    </revision>
    <revision>
      <id>2834726</id>
      <parentid>2834688</parentid>
      <timestamp>2026-09-27T22:00:08Z</timestamp>
      <contributor>
        <username>~2026-52080-44</username>
        <id>3111727</id>
      </contributor>
      <comment>/* I am trying to publish a biography */ Reply</comment>
      <origin>2834726</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10335" sha1="tdquozbdcf7kr41sj1yx25uaza4sqxu" xml:space="preserve">{{/Header}}

==Review changes to [[Special:AbuseFilter/4]]==
{{ping|Codename Noreste}} Could you review changes I made to this filter to help prevent profanity spam using obfuscated spellings? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:44, 25 May 2026 (UTC)
: I've adjusted the filter to prevent any potential false positives, see its conditions. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:10, 26 May 2026 (UTC)

== Please unban ==

I want to create a new page (a pro/contra discussion page [[Are humans omnivores or herbivores?|like this one]]); initially it included an entry in the &quot;== External Links ==&quot; section, so it got blocked. I tried using a doi.org link; unfortunately that got blocked, too. Then I tried without, and it said I tried too often…
Please unban my user so I can create the page (without an external link…). In general, please consider adding doi.org to a whitelist. DOI links are virtually all academic, so how bad can that possibly be?
Thank you! :-) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:02, 15 August 2026 (UTC)
: Thanks for letting us know. Some suggestion:
:* Although you can edit anonymously, it is better to create an account and log in - that will give you more permissions
:* You are hitting some spam filters, but the system will learn to trust you if you login and engage in some constructive editing (e.g., edit and improve a page before before adding external links); basically, it doesn't trust you yet. We don't have a lot of volunteer admins so it isn't perfect (there are some false positives), but it prevents a lot of clean-up.
: 
: But now for the not-so-good news. The Wikiversity community recently decide to stop Wikidebates because there were some problematic topics and responses. However, if it is a good topic, there's likely some way to do something similar that would be accepted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 15 August 2026 (UTC)
::My debate question is: &quot;Are humans monogamous or polygamous?&quot; (In my opinion, that makes it very analogous to the diet question I liked to.) [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 11:37, 15 August 2026 (UTC)
::: This question is an example of a potentially educationally relevant topic to have as a learning project in debate format, but currently debates are not permitted (see [[Wikidebates]] for more info). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:03, 15 August 2026 (UTC)
::::I understand. I would generally find it noteworthy that a debate club which used to pretend to be liberal would so readily admit to the opposite and shut itself down, simply because – I'm assuming – someone asked the forbidden questions of our time or questioned the truisms of our time, which stirred up some emotions in the overly sensitive. That, of course, is not what the enlightenment taught us, nor is it in line with what ''actually'' liberal academia is all about. However, it is no surprise to me at all for a socialist project of the Wikimedia realm. I'll go and find myself some better people. Thank you and have a good day, Jtneill. [[Special:Contributions/&amp;#126;2026-44661-47|&amp;#126;2026-44661-47]] ([[User talk:&amp;#126;2026-44661-47|talk]]) 12:10, 15 August 2026 (UTC)

== Request to review AbuseFilter block on a student research draft ==

Hello. I am a new contributor attempting to create [[Draft:Anti-IL-6 scFv–RDVLPGT fusion concept]], a clearly labeled student research project. The draft separates cited evidence from an unvalidated design hypothesis, states the central limitation (extracellular anti-IL-6 scFv versus intracellular MyD88), includes falsifiable expectations, and uses external links only for DOI references, RCSB PDB 4DOM, and an NCBI iCn3D structural-context view. The edit was disallowed by the rule “New user creating page with external link.” Could a custodian please review the draft and allow the constructive edit, or advise the appropriate way to publish it without removing its verifiable sources? Thank you. [[User:Jayhan0302|Jayhan0302]] ([[User talk:Jayhan0302|discuss]] • [[Special:Contributions/Jayhan0302|contribs]]) 08:47, 22 August 2026 (UTC)
: Hi Jayhan,
: Because you're a new user, the system wants to learn to trust you, so try editing at first without including external links. Then, after a while, it will realise that you're to be trusted, and then you'll be able to add external links.
: I hope that helps.
: Sincerely,&lt;br&gt; James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:04, 22 August 2026 (UTC)
:This is a little embarrassing, but I've never been an EditFilter whiz, so I just added the links myself. You will be able to add links yourself after you have made &lt;var&gt;x&lt;/var&gt; edits in &lt;var&gt;y&lt;/var&gt; days. I can never recall the exact numbers, but they are very small. Let me know if you need more help. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:06, 22 August 2026 (UTC)

== Report concerning [[special:contribs/Bśadów$ki112|Bśadów$ki112]] ==
'''Reasons''': Long-term abuse. Impersonation of [[:sulutil:Bsadowski1|Bsadowski1]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 19:19, 28 August 2026 (UTC)

:{{done}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 19:20, 28 August 2026 (UTC)

== Custodianship ==
proofreading and summarizing I excel in  social science and current events history political science are my areas of past study. Could I be custodian of primarily pages dealing with social science and politics? I have experience editing am a advanced reader and can accept more than one point of view on almost any topic the point is to see the ethical standard and protection of people of sacred belief of food equitable relationships and not accept any point of view that includes towards a laissez faire attitude on abuse. To me it's what we can do as individuals,as community,as friends as family not what harm is possible for that is off the boards not acceptable. I'll keep a open mind and not stifle any dissent. My interest is also in assuring the complete construction of explanatory paragraphs,proper wording and generally good punctuation.  [[Special:Contributions/~2026-50875-97|~2026-50875-97]] ([[User talk:~2026-50875-97|talk]]) 15:27, 21 September 2026 (UTC)

:What you posted here does not require custodian action and show some poor judgement: This board is for escalated issues that a standard user can't perform. What you are basically writing here is &quot;I would like to be a custodian&quot;. There is a process for that, but step one would be having an actual account to edit from, so do that first, then edit here as a non-custodian in a helpful manner, and then make the request if you'd still like that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:38, 21 September 2026 (UTC)

== I am trying to publish a biography ==
I am a beginner, I cannot succeed. [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 12:57, 27 September 2026 (UTC)

:I posted a [[Template:welcome|welcome message]] to your talk page that gives you an overview of Wikiversity. While a biography &lt;em&gt;could&lt;/em&gt; be considered appropriate here if part of a larger exercise in learning but this is not an encyclopedia like our sibling project Wikipedia, so generally, we won't have much in the way of article-long biographies. Why were you trying to post a biography here and of whom? ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:24, 27 September 2026 (UTC)
::I am a writer I publish in Romania and USA, and I wanted to put on-line officially, the Bibliography, both in romanian and english, maybe I am not on the right topic pages, I did much work in Romania, and both in USA, and I am proud, but just maybe I didn't came out with the right issued page, I will do my best to make a bibliography, if I may. It was about my works, so far. I will try another day, if I succeed... Very much greetings Iulia, ro Where I can put the text, I have seen other Bibliographies, on wiki? [[User:Petrosa51|Petrosa51]] ([[User talk:Petrosa51|discuss]] • [[Special:Contributions/Petrosa51|contribs]]) 16:41, 27 September 2026 (UTC)
:::If you want to contribute here and you want to mention your publications on your userpage, I think that's fine. If you just want to make a user page for vanity's sake and aren't going to actually contribute here, I don't think that's useful: we don't provide free hosting. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:11, 27 September 2026 (UTC)
::::Thank you very much for your response, I will agree with that. I have seen book lists at other authors, and that was why, is not vanity is work, Iulia E., from Romania... (Wikipedia hosted that kind, it was not my invention), maybe I will contribute if I may with other things, with much friendship, Iulia [[Special:Contributions/~2026-52080-44|~2026-52080-44]] ([[User talk:~2026-52080-44|talk]]) 22:00, 27 September 2026 (UTC)</text>
      <sha1>tdquozbdcf7kr41sj1yx25uaza4sqxu</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Lectures/Aspects of emotion</title>
    <ns>0</ns>
    <id>98607</id>
    <revision>
      <id>2834796</id>
      <parentid>2833250</parentid>
      <timestamp>2026-09-28T04:54:42Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Readings */ Update for 2026</comment>
      <origin>2834796</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12027" sha1="0b8sbixis1ik27wob1e25a90cnl65u8" xml:space="preserve">{{Motivation and emotion/Lectures|Lecture 8: Aspects of emotion|eighth}}
{{Motivation and emotion/Lectures/In development}}
&lt;!-- {{Motivation and emotion/Lectures/In development}} --&gt;
&lt;!-- {{Motivation and emotion/Lectures/Complete}} --&gt;
==Overview==
[[File:Two people with brain cogs turning.png|right|250px|thumb|&lt;div style='text-align: center;'&gt;What causes emotion?&lt;br&gt;Biology?&lt;br&gt;Cognition?&lt;br&gt;Sociality?&lt;/div&gt;]]

This lecture:
* discusses biological, cognitive, and social psychological aspects of [[emotion]] 
* considers the potential of [[w:affective computing|affective computing]]

Take-home messages:
* Biology and cognition work together to generate emotion
* Facial expressions can influence emotional state
* Other people are the source of most of our emotions
* Emotions are contagious
* Computers are increasingly able to detect, imitate, and respond to emotion

==Outline==
* Biological
* Cognitive
* Social
* Affective computing

==Biological aspects of emotion==
[[File:Bipolar Dyptych 1 365.jpg|thumb|right|200px|'''Figure 1'''. What is the role of the body in causing emotion?]]
[[File:Expression of the Emotions Figure 1.png|thumb|right|200px|'''Figure 2'''. The human face has over 80 muscles; ~half of which are involved in expressing ~3,000 different expressions of emotion]]
* What is the role of the body in emotion (see Figure 1)?
** Does bodily reaction follow emotion?
** Or does bodily reaction lead to emotion?
* [[w:James–Lange theory|James–Lange theory of emotion]]:
** Sequence
*** Stimulus → Bodily reaction → Emotion e.g.,
*** Sudden cold shower → Increased heart rate → Surprise? Fear?
** Emotion is a way of making sense of bodily changes
** Criticisms
*** Boldy reactions are part of a general response that does not vary much between emotions
*** Emotional experiences occur more quickly than physiological reactions
** Contemporary perspective
*** Distinct physiological differences are evident for some emotions, but only a few have distinct [[w:Autonomic nervous system|autonomic nervous system]] patterns
*** Emotions recruit bodily reactions to facilitate adaptative behaviours such as fighting, fleeing, and nurturing
* Brain activations for specific emotions
** Distinct neural circuits underlie the core emotions of joy, fear, rage, and anxiety ([[w:Gray's biopsychological theory of personality|Gray]])
*** Behavioural approach system
*** Fight or flight system
*** Behavioural inhibition system
** Neuroscience studies of brain activity during emotional experiences map core emotions to distinct patterns of neural activity
* [[w:Facial feedback hypothesis|Facial feedback hypothesis]]
** The facial feedback hypothesis (FFH) proposes that emotional experience arises from interpretation of one's facial expression
*** Strong view: FF causes emotion
*** Weak view: FF modifies emotion
*** Critics: FF effect is small
** Facial expression of core emotions is cross-culturally universal (see Figure 2)
*** Some emotions (e.g., joy) are easier to recognise than others (e.g., fear)
*** Some cultures (e.g., Western) are better at recognising emotion (e.g., because they focus on the mouth) than other cultures (e.g., Eastern because they focus on the eyes)

==Cognitive aspects of emotion==
* Biology alone doesn't explain all aspects of emotion, particularly complex emotions such as hope, pride, envy, and gratitude
* Appraisal
** Appraisals, rather than events per se, elicit emotion
** Appraisals evaluates significance of events
*** Primary: &quot;Is this event significant to me and my well-being?&quot;; if yes, &quot;Is the event good (beneficial) or bad (harmful)?&quot;
*** Secondary: &quot;Can I cope with this situation?&quot; → emotion (e.g., liking/disliking) → approach vs. withdrawal
** Complex appraisal models account for approx. 2/3rds of emotional variability by mapping perceived types of benefit/harm/threat to specific emotions
* Emotion knowledge
** Ability to differentiate different types and intensities of emotion
** Part of [[w:Emotional intelligence|emotional intelligence]]
** Can be taught and developed
* Attribution
** Different explanations for why life events occur leads to different emotional experiences
*** Primary: Good or bad?
*** Secondary: Cause?
** Example:
*** Primary: Good event
*** Secondary: Internal cause
*** Emotion: Pride

==Social aspects of emotion==
[[File:Just love cropped cropped.jpg|thumb|right|200px|'''Figure 3'''. Emotions are socially intense experiences, bringing us together and driving us apart]]
* Emotions are socially intense experiences (see Figure 3)
** Through emotional expression, we signal our needs to others
** Social experiences are the most frequent source of day-to-day emotion; these experiences:
*** bring us together (e.g., joy and gratitude) and 
*** push us apart (e.g., anger, contempt, and schadenfreude)
** [[w:Emotional contagion|Emotional contagion]]: People tend to converge on similar emotions due to:
*** [[w:Mirror neuron|Mirror neuron]]s fire in response to observing others' emotion
*** People tend to mimic others' emotional expressions
* Social sharing of emotion provides conversational context for people to re-experience and recount their emotional experiences
** Social-affective sharing: Listening, comforting, empathy, support → provides temporary relief
** Cognitive sharing: Reframing, meaning-making, reprioritising → stimulates cognitive work for emotional healing and recovery
* Common view of emotions as short-lived is challenged by social research about emotion because emotional experiences tend to be retold and relived through social sharing, contributing to emotional continuity over days, weeks, and even years

==Affective computing==
[[File:Sophia (robot).jpg|thumb|200px|'''Figure 4'''. Electronic devices such as robots are increasingly able to detect and respond to human emotion.]]
* [[w:Affective computing|Affective computing]] is an interdisciplinary field which studies how [[w:Artificial intelligence|artificial intelligence]] can recognise and respond to human emotion
* If emotions show ANS specificity, then sensors built into electronic devices can monitor human and adjust accordingly (e.g., empathically) (see Figure 4)

==Readings==
* Chapter 12: Aspects of emotion ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]])

==Multimedia==
* [https://www.youtube.com/embed/-PFqzYoKkCc?si=e9oy7d30rVETd82h&amp;amp;start=12&amp;end=679 Emotions revealed] (KQED QUEST, 2008, YouTube) (11:01 mins): Explains Paul Ekman's work on codifying the intricate ways in which emotions are revealed through facial expressions
* [https://www.youtube.com/embed/TdsFGqhoAEo?si=slTlN00GwXbELkDM&amp;amp;start=18&amp;end=160 Why we can't not smile] (Epic Science, 2014, YouTube) (2:22 mins): Explores unconscious emotional mimicry in relation to smiling
* [https://www.youtube.com/watch?v=JMLsHI8aV0g How China is using artificial intelligence in classrooms] (The Wall Street Journal, 2020, YouTube) (5:43 mins): Shows real-world application of emotion AI being used to monitor students, illustrating potentials and risks of affective computing

==Slides==
* [https://docs.google.com/presentation/d/1b6mSQQGASDgImEPjDt9DyU4Tm2NtT1GO1k0UdWLSSeE/edit?usp=sharing Aspects of emotion] (Google Slides)
&lt;!-- * [http://www.slideshare.net/jtneill/aspects-of-emotion Lecture slides] (Slideshare)
* Handouts
** [[Media:Motivation and Emotion - Lecture 08 - Aspects of emotion 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 08 - Aspects of emotion 6slidesperpage.pdf|100px]]
** [[Media:Motivation and Emotion - Lecture 08 - Aspects of emotion 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 08 - Aspects of emotion 3slidesperpage.pdf|100px]]
* [http://www.slideshare.net/jtneill/aspects-of-emotion Lecture slides] (Slideshare)
--&gt;

==See also==
* [[/Images/]]

; Lectures
* [[{{#titleparts:{{PAGENAME}}|2}}/Nature of emotion|Nature of emotion]] (Previous lecture)
* [[{{#titleparts:{{PAGENAME}}|2}}/Individual emotions|Individual emotions]] (Next lecture)

;Tutorial
* [[{{#titleparts:{{PAGENAME}}|1}}/Tutorials/Measuring emotion|Measuring emotion]] (Tutorial)

;Wikiversity
* [[:Category:Motivation and emotion/Book/Affective computing|Affective computing]] (Book chapter category)
* [[Motivation and emotion/Book/2014/Appraisal and emotion|Appraisal and emotion]] (Book chapter, 2014)
&lt;!-- [[Motivation and emotion/Book/2024/Attribution theory and emotion|Attribution theory and emotion]] (Book chapter 2024) --&gt;
* [[Motivation and emotion/Book/2014/Facial Action Coding System|Facial Action Coding System]] (Book chapter, 2014)
* [[Motivation and emotion/Book/2023/Smiling and emotion|Smiling and emotion]] (Book chapter, 2023)

;Wikipedia

Topics
* [[w:Affective computing|Affective computing]]
* [[w:Appraisal theory|Appraisal]]
* [[w:Attribution (psychology)|Attribution]]
* [[w:Gray's biopsychological theory of personality|Gray's biopsychological theory of personality]]
* [[w:James–Lange theory|James–Lange theory]]
* [[w:Sophia (robot)|Sophia (robot)]]

People
* [[w:Magda B. Arnold|Magda B. Arnold]]
* [[w:Paul Ekman|Paul Ekman]]
* [[w:Jeffrey Alan Gray|Jeffrey Gray]]
* [[w:Carroll Izard|Carroll Izard]]

==Recording==
* [Lecture 8] (2026)
* [https://au-lti.bbcollab.com/recording/91c30feb657c4f0db225b3babaefd962 Lecture 8] (2025)&lt;!--
* [https://au-lti.bbcollab.com/recording/9c2e9707be814a9d87cc9131d0782390 Lecture 8 recording] (2024)
* [https://au-lti.bbcollab.com/recording/4c3533d7c9294b2da8dc4aaa8aac0b93 Lecture 8 recording] (2023)
* [https://au-lti.bbcollab.com/recording/def61ff43cfb4f0084e26aac0bc0b52f Lecture 8 recording] (2022)
* [https://au-lti.bbcollab.com/recording/7a2b4dd5a81246c096581c3461eae997 Lecture 8 recording] (2021)
* [https://au-lti.bbcollab.com/recording/9173da24929249c0905c62f676287d83 Lecture 8 recording] (2020)
* [https://echo360.org.au/media/00c6015a-2dd9-48c5-9d78-1f1891908d30/public Lecture 8 recording] (2019)
* [https://echo360.org.au/media/17d036bf-4c49-45a9-9dd7-6d9364a51263/public Lecture 8 recording] (2016) --&gt;

==References==
{{Hanging indent|1=

Gray, J. A. (1987). ''The psychology of fear and stress'' (2nd ed.). Cambridge University Press.
&lt;!-- 
Izard, C. E. (2010). The many meanings/aspects of emotion: Definitions, functions, activation, and regulation. ''Emotion Review'', ''2''(4), 363–370. https://doi.org/10.1177/1754073910374661
--&gt;
}}

==External links==
* [https://www.paulekman.com Paul Ekman Group] (paulekman.com)
* [https://www.bps.org.uk/research-digest/precise-meaning-emotion-words-different-around-world The precise meaning of emotion words is different around the world] (bps.org.uk)
* [https://www.youtube.com/watch?v=2Qb_XnEUcY8 The scary truth behind AI's emotional power] (AI for Humans, 2025; YouTube 11:42 mins)

;Multimedia
* [https://www.youtube.com/watch?v=ui2_mZWyYK8 Can humanoid robots like Ameca develop emotions?] (Genz Ai Revolution, 2025, YouTube) (3:13 mins)&lt;!-- : Explores whether robots genuinely feel emotions or merely simulate them through sensors and advanced programming. Raises ethical questions about human–robot interaction, emotional mimicry, and the implications of machines acting as companions, caretakers, or friends. --&gt;
* [https://youtu.be/WY0j1cZtnp0 In your face] (Mind Field Ep 7, YouTube, 24:33 mins)
* [https://www.youtube.com/watch?v=Sq36J9pNaEo Meet Sophia, World's first AI humanoid robot] (Tony Robbins, 2020, YouTube; 9:55 mins): Interview with an AI robot about life, the universe, and everything
* [https://www.youtube.com/watch?v=wjW-v0IPT_M Robots that show emotion] (David Hanson, TED talk, 7:29 mins)
* [https://www.youtube.com/watch?v=U9cGdRNMdQQ The hidden power of smiling] (Ron Gutman, TED talk; 7:26 mins)
* [https://www.youtube.com/watch?v=Uq6XgrYBugo Why we have an emotional connection to robots] (Kate Darling, 2018, TED talk; 11:51 mins)
{{Motivation and emotion/Lectures/Navigation}}

[[Category:Motivation and emotion/Lectures/Aspects of emotion]]</text>
      <sha1>0b8sbixis1ik27wob1e25a90cnl65u8</sha1>
    </revision>
    <revision>
      <id>2834798</id>
      <parentid>2834796</parentid>
      <timestamp>2026-09-28T04:57:10Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Update for 2026</comment>
      <origin>2834798</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12027" sha1="385frfu5912jqk1oc3ng6jvd5apuygi" xml:space="preserve">{{Motivation and emotion/Lectures|Lecture 8: Aspects of emotion|eighth}}
{{Motivation and emotion/Lectures/Complete}}
&lt;!-- {{Motivation and emotion/Lectures/In development}} --&gt;
&lt;!-- {{Motivation and emotion/Lectures/Complete}} --&gt;
==Overview==
[[File:Two people with brain cogs turning.png|right|250px|thumb|&lt;div style='text-align: center;'&gt;What causes emotion?&lt;br&gt;Biology?&lt;br&gt;Cognition?&lt;br&gt;Sociality?&lt;/div&gt;]]

This lecture:
* discusses biological, cognitive, and social psychological aspects of [[emotion]] 
* considers the potential of [[w:affective computing|affective computing]]

Take-home messages:
* Biology and cognition work together to generate emotion
* Facial expressions can influence emotional state
* Other people are the source of most of our emotions
* Emotions are contagious
* Computers are increasingly able to detect, imitate, and respond to human emotion

==Outline==
* Biological
* Cognitive
* Social
* Affective computing

==Biological aspects of emotion==
[[File:Bipolar Dyptych 1 365.jpg|thumb|right|200px|'''Figure 1'''. What is the role of the body in causing emotion?]]
[[File:Expression of the Emotions Figure 1.png|thumb|right|200px|'''Figure 2'''. The human face has over 80 muscles; ~half of which are involved in expressing ~3,000 different expressions of emotion]]
* What is the role of the body in emotion (see Figure 1)?
** Does bodily reaction follow emotion?
** Or does bodily reaction lead to emotion?
* [[w:James–Lange theory|James–Lange theory of emotion]]:
** Sequence
*** Stimulus → Bodily reaction → Emotion e.g.,
*** Sudden cold shower → Increased heart rate → Surprise? Fear?
** Emotion is a way of making sense of bodily changes
** Criticisms
*** Boldy reactions are part of a general response that does not vary much between emotions
*** Emotional experiences occur more quickly than physiological reactions
** Contemporary perspective
*** Distinct physiological differences are evident for some emotions, but only a few have distinct [[w:Autonomic nervous system|autonomic nervous system]] patterns
*** Emotions recruit bodily reactions to facilitate adaptative behaviours such as fighting, fleeing, and nurturing
* Brain activations for specific emotions
** Distinct neural circuits underlie the core emotions of joy, fear, rage, and anxiety ([[w:Gray's biopsychological theory of personality|Gray]])
*** Behavioural approach system
*** Fight or flight system
*** Behavioural inhibition system
** Neuroscience studies of brain activity during emotional experiences map core emotions to distinct patterns of neural activity
* [[w:Facial feedback hypothesis|Facial feedback hypothesis]]
** The facial feedback hypothesis (FFH) proposes that emotional experience arises from interpretation of one's facial expression
*** Strong view: FF causes emotion
*** Weak view: FF modifies emotion
*** Critics: FF effect is small
** Facial expression of core emotions is cross-culturally universal (see Figure 2)
*** Some emotions (e.g., joy) are easier to recognise than others (e.g., fear)
*** Some cultures (e.g., Western) are better at recognising emotion (e.g., because they focus on the mouth) than other cultures (e.g., Eastern because they focus on the eyes)

==Cognitive aspects of emotion==
* Biology alone doesn't explain all aspects of emotion, particularly complex emotions such as hope, pride, envy, and gratitude
* Appraisal
** Appraisals, rather than events per se, elicit emotion
** Appraisals evaluates significance of events
*** Primary: &quot;Is this event significant to me and my well-being?&quot;; if yes, &quot;Is the event good (beneficial) or bad (harmful)?&quot;
*** Secondary: &quot;Can I cope with this situation?&quot; → emotion (e.g., liking/disliking) → approach vs. withdrawal
** Complex appraisal models account for approx. 2/3rds of emotional variability by mapping perceived types of benefit/harm/threat to specific emotions
* Emotion knowledge
** Ability to differentiate different types and intensities of emotion
** Part of [[w:Emotional intelligence|emotional intelligence]]
** Can be taught and developed
* Attribution
** Different explanations for why life events occur leads to different emotional experiences
*** Primary: Good or bad?
*** Secondary: Cause?
** Example:
*** Primary: Good event
*** Secondary: Internal cause
*** Emotion: Pride

==Social aspects of emotion==
[[File:Just love cropped cropped.jpg|thumb|right|200px|'''Figure 3'''. Emotions are socially intense experiences, bringing us together and driving us apart]]
* Emotions are socially intense experiences (see Figure 3)
** Through emotional expression, we signal our needs to others
** Social experiences are the most frequent source of day-to-day emotion; these experiences:
*** bring us together (e.g., joy and gratitude) and 
*** push us apart (e.g., anger, contempt, and schadenfreude)
** [[w:Emotional contagion|Emotional contagion]]: People tend to converge on similar emotions due to:
*** [[w:Mirror neuron|Mirror neuron]]s fire in response to observing others' emotion
*** People tend to mimic others' emotional expressions
* Social sharing of emotion provides conversational context for people to re-experience and recount their emotional experiences
** Social-affective sharing: Listening, comforting, empathy, support → provides temporary relief
** Cognitive sharing: Reframing, meaning-making, reprioritising → stimulates cognitive work for emotional healing and recovery
* Common view of emotions as short-lived is challenged by social research about emotion because emotional experiences tend to be retold and relived through social sharing, contributing to emotional continuity over days, weeks, and even years

==Affective computing==
[[File:Sophia (robot).jpg|thumb|200px|'''Figure 4'''. Electronic devices such as robots are increasingly able to detect and respond to human emotion.]]
* [[w:Affective computing|Affective computing]] is an interdisciplinary field which studies how [[w:Artificial intelligence|artificial intelligence]] can recognise and respond to human emotion
* If emotions show ANS specificity, then sensors built into electronic devices can monitor human and adjust accordingly (e.g., empathically) (see Figure 4)

==Readings==
* Chapter 12: Aspects of emotion ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]])

==Multimedia==
* [https://www.youtube.com/embed/-PFqzYoKkCc?si=e9oy7d30rVETd82h&amp;amp;start=12&amp;end=679 Emotions revealed] (KQED QUEST, 2008, YouTube) (11:01 mins): Explains Paul Ekman's work on codifying the intricate ways in which emotions are revealed through facial expressions
* [https://www.youtube.com/embed/TdsFGqhoAEo?si=slTlN00GwXbELkDM&amp;amp;start=18&amp;end=160 Why we can't not smile] (Epic Science, 2014, YouTube) (2:22 mins): Explores unconscious emotional mimicry in relation to smiling
* [https://www.youtube.com/watch?v=JMLsHI8aV0g How China is using artificial intelligence in classrooms] (The Wall Street Journal, 2020, YouTube) (5:43 mins): Shows real-world application of emotion AI being used to monitor students, illustrating potentials and risks of affective computing

==Slides==
* [https://docs.google.com/presentation/d/1b6mSQQGASDgImEPjDt9DyU4Tm2NtT1GO1k0UdWLSSeE/edit?usp=sharing Aspects of emotion] (Google Slides)
&lt;!-- * [http://www.slideshare.net/jtneill/aspects-of-emotion Lecture slides] (Slideshare)
* Handouts
** [[Media:Motivation and Emotion - Lecture 08 - Aspects of emotion 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 08 - Aspects of emotion 6slidesperpage.pdf|100px]]
** [[Media:Motivation and Emotion - Lecture 08 - Aspects of emotion 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 08 - Aspects of emotion 3slidesperpage.pdf|100px]]
* [http://www.slideshare.net/jtneill/aspects-of-emotion Lecture slides] (Slideshare)
--&gt;

==See also==
* [[/Images/]]

; Lectures
* [[{{#titleparts:{{PAGENAME}}|2}}/Nature of emotion|Nature of emotion]] (Previous lecture)
* [[{{#titleparts:{{PAGENAME}}|2}}/Individual emotions|Individual emotions]] (Next lecture)

;Tutorial
* [[{{#titleparts:{{PAGENAME}}|1}}/Tutorials/Measuring emotion|Measuring emotion]] (Tutorial)

;Wikiversity
* [[:Category:Motivation and emotion/Book/Affective computing|Affective computing]] (Book chapter category)
* [[Motivation and emotion/Book/2014/Appraisal and emotion|Appraisal and emotion]] (Book chapter, 2014)
&lt;!-- [[Motivation and emotion/Book/2024/Attribution theory and emotion|Attribution theory and emotion]] (Book chapter 2024) --&gt;
* [[Motivation and emotion/Book/2014/Facial Action Coding System|Facial Action Coding System]] (Book chapter, 2014)
* [[Motivation and emotion/Book/2023/Smiling and emotion|Smiling and emotion]] (Book chapter, 2023)

;Wikipedia

Topics
* [[w:Affective computing|Affective computing]]
* [[w:Appraisal theory|Appraisal]]
* [[w:Attribution (psychology)|Attribution]]
* [[w:Gray's biopsychological theory of personality|Gray's biopsychological theory of personality]]
* [[w:James–Lange theory|James–Lange theory]]
* [[w:Sophia (robot)|Sophia (robot)]]

People
* [[w:Magda B. Arnold|Magda B. Arnold]]
* [[w:Paul Ekman|Paul Ekman]]
* [[w:Jeffrey Alan Gray|Jeffrey Gray]]
* [[w:Carroll Izard|Carroll Izard]]

==Recording==
* [Lecture 8] (2026)
* [https://au-lti.bbcollab.com/recording/91c30feb657c4f0db225b3babaefd962 Lecture 8] (2025)&lt;!--
* [https://au-lti.bbcollab.com/recording/9c2e9707be814a9d87cc9131d0782390 Lecture 8 recording] (2024)
* [https://au-lti.bbcollab.com/recording/4c3533d7c9294b2da8dc4aaa8aac0b93 Lecture 8 recording] (2023)
* [https://au-lti.bbcollab.com/recording/def61ff43cfb4f0084e26aac0bc0b52f Lecture 8 recording] (2022)
* [https://au-lti.bbcollab.com/recording/7a2b4dd5a81246c096581c3461eae997 Lecture 8 recording] (2021)
* [https://au-lti.bbcollab.com/recording/9173da24929249c0905c62f676287d83 Lecture 8 recording] (2020)
* [https://echo360.org.au/media/00c6015a-2dd9-48c5-9d78-1f1891908d30/public Lecture 8 recording] (2019)
* [https://echo360.org.au/media/17d036bf-4c49-45a9-9dd7-6d9364a51263/public Lecture 8 recording] (2016) --&gt;

==References==
{{Hanging indent|1=

Gray, J. A. (1987). ''The psychology of fear and stress'' (2nd ed.). Cambridge University Press.
&lt;!-- 
Izard, C. E. (2010). The many meanings/aspects of emotion: Definitions, functions, activation, and regulation. ''Emotion Review'', ''2''(4), 363–370. https://doi.org/10.1177/1754073910374661
--&gt;
}}

==External links==
* [https://www.paulekman.com Paul Ekman Group] (paulekman.com)
* [https://www.bps.org.uk/research-digest/precise-meaning-emotion-words-different-around-world The precise meaning of emotion words is different around the world] (bps.org.uk)
* [https://www.youtube.com/watch?v=2Qb_XnEUcY8 The scary truth behind AI's emotional power] (AI for Humans, 2025; YouTube 11:42 mins)

;Multimedia
* [https://www.youtube.com/watch?v=ui2_mZWyYK8 Can humanoid robots like Ameca develop emotions?] (Genz Ai Revolution, 2025, YouTube) (3:13 mins)&lt;!-- : Explores whether robots genuinely feel emotions or merely simulate them through sensors and advanced programming. Raises ethical questions about human–robot interaction, emotional mimicry, and the implications of machines acting as companions, caretakers, or friends. --&gt;
* [https://youtu.be/WY0j1cZtnp0 In your face] (Mind Field Ep 7, YouTube, 24:33 mins)
* [https://www.youtube.com/watch?v=Sq36J9pNaEo Meet Sophia, World's first AI humanoid robot] (Tony Robbins, 2020, YouTube; 9:55 mins): Interview with an AI robot about life, the universe, and everything
* [https://www.youtube.com/watch?v=wjW-v0IPT_M Robots that show emotion] (David Hanson, TED talk, 7:29 mins)
* [https://www.youtube.com/watch?v=U9cGdRNMdQQ The hidden power of smiling] (Ron Gutman, TED talk; 7:26 mins)
* [https://www.youtube.com/watch?v=Uq6XgrYBugo Why we have an emotional connection to robots] (Kate Darling, 2018, TED talk; 11:51 mins)
{{Motivation and emotion/Lectures/Navigation}}

[[Category:Motivation and emotion/Lectures/Aspects of emotion]]</text>
      <sha1>385frfu5912jqk1oc3ng6jvd5apuygi</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Newsletters/Tech News</title>
    <ns>4</ns>
    <id>162205</id>
    <revision>
      <id>2834833</id>
      <parentid>2834670</parentid>
      <timestamp>2026-09-28T10:49:10Z</timestamp>
      <contributor>
        <username>MediaWiki message delivery</username>
        <id>983498</id>
      </contributor>
      <comment>/* Tech News: 2026-40 */ new section</comment>
      <origin>2834833</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="53314" sha1="bu8q2bhifx51w5rbdhfu3nk99xzc8wj" xml:space="preserve">{{Archive box|[[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]] · [[/2022/]] · [[/2023/]] · [[/2024/]] · [[/2025/]]}}
__TOC__
{{Clear}}

== Tech News: 2026-31 ==

&lt;section begin=&quot;technews-2026-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/31|Translations]] are available.

'''Updates for editors'''
* [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] [[mw:Special:MyLanguage/ContentTranslation|Content Translation]] now supports dark mode, fulfilling a [[m:Community Wishlist/W544|Community Wishlist request]]. This brings the tool in line with the accessibility features available in the Vector 2022 and Minerva skins, helping reduce visual fatigue for users translating content. [https://phabricator.wikimedia.org/T367077]
* DiscussionTools' source mode and the 2017 wikitext editor will now offer autocomplete for links (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;[[&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), templates (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), HTML and parser tags (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), and magic words (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), making it quicker and easier to insert links, templates, and other wiki markup while editing. [https://phabricator.wikimedia.org/T432400]
* The [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews|Readers Growth team]] has concluded its experiment with mobile page previews and will not roll out the feature. Page Previews are a pop-up bottom sheet that appears when readers tap a blue link, showing a thumbnail, lead paragraph, and an option to open the article. The experiment showed flat retention and negative indicator metrics, suggesting that mobile web readers preferred navigating directly to linked articles rather than using page previews.
* The Reader Experience team has seen encouraging early results from the [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists feature]], with 93% of participating users reporting that it was useful. Reading Lists help active readers save articles for future reading and support their learning goals on Wikimedia projects. The team plans further improvements before expanding the feature to more users.
* The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Explore Feed Refresh|Explore Feed Refresh]] initiative was tested with new and casual Wikipedia app readers. The refreshed feed helps readers discover new and relevant content. After a 10.5% increase in engagement with the feed, Wikimedia Apps team has decided to scale the Home Feed redesign to iOS with the learnings from the Android release applied.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where subject names in the Article Guidance feature were displayed with incorrect capitalization on French Wikipedia, has now been fixed. Subject names will now follow the correct capitalization rules for the language. [https://phabricator.wikimedia.org/T427201]

'''Updates for technical contributors'''
* After running several [[mw:Special:MyLanguage/Contributors/Account Creation Experiments|Account Creation Experiments]] to improve registration completion rates, a new version of the username field on [[Special:CreateAccount|Create Account]] has been rolled out. It includes [[:c:File:Create account - July 2026 updates.png|a popover summarizing the username policy]] to provide clearer guidance during account creation. As part of this change, the messages &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-helpusername&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-help&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; that several communities have configured will no longer be used. If communities want to customize the guidance shown in the new popover, they can instead edit the following messages: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-policy-popover-bullet1&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-policy-popover-bullet2&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-policy-popover-bullet3&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T430604]
* Later this week, the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|CodeMirror syntax highlighter]] will offer [[w:en:Theme (computing)|themes]]. The themes can be picked from a dropdown menu in the full [[mw:Special:MyLanguage/Help:Extension:CodeMirror#CodeMirror preferences|CodeMirror preferences]] dialog. For wikitext, available themes are default, colorblind-friendly (previously the colorblind preference option on [[Special:Preferences#mw-prefsection-editing]]) and no-highlighting. For code languages (i.e., CSS/JavaScript/JSON/Vue/Lua), there are several themes available. These same themes will eventually be available for wikitext, too. [https://phabricator.wikimedia.org/T163533]
* From now on, wikis can restrict editing in the &quot;User&quot; namespace to only the page owner and certain user groups. [[mw:Special:MyLanguage/Manual:$wgRestrictUserPageEditing|Read the configuration documentation]] to learn more.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.14|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:49, 27 July 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30856308 --&gt;

== Tech News: 2026-32 ==

&lt;section begin=&quot;technews-2026-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/32|Translations]] are available.

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] has developed a [https://82db7c8d4b.catalyst.wmcloud.org/w/index.php?title=Regent%27s_Park&amp;uselang=de patch demo] that wraps the page toolbar onto two lines when there is not enough horizontal space for all the buttons. This aims to reduce crowding in the Vector 2022 toolbar, which can occur on some language Wikipedias at certain screen widths. [https://phabricator.wikimedia.org/T429518]
* The Reader Experience team is planning to launch [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists]], a [[m:Special:MyLanguage/Community Wishlist Survey 2021/Mobile and apps/Have Apps reading lists available on Destop/Mobile|Community Wishlist item]], which is currently available to try in beta, as a full feature in September. Before then, volunteer translator help is needed for [https://translatewiki.net/w/i.php?title=Special%3ATranslate&amp;group=ext-readinglists&amp;filter=&amp;action=translate string translations] into a number of languages. The feature supports reading and learning goals on Wikipedia.
* Next week, the table of contents on Wikimedia Commons file pages will be improved by consolidating the file page table of contents with the page table of contents. This will make it easier to understand a file page’s structure, navigate to specific sections, and share links to individual sections. [https://phabricator.wikimedia.org/T332644]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where some [[w:TIFF|TIFF]] images failed to load after clicking their thumbnail, causing a broken image to be displayed instead of the full-size image, has now been fixed. [https://phabricator.wikimedia.org/T429326]

'''Updates for technical contributors'''
* The variable and function selector in AbuseFilter has been updated to support search and autocomplete. It will allow filter maintainers to find the desired variable or function more quickly. [https://phabricator.wikimedia.org/T323698]
* The MJPEG and VP8 formats are removed from the video player. The MP4 format (MPEG-4 Part 2) is added instead, which provides higher quality videos to older iPhone devices. It may take a few weeks to retroactively update all existing videos. The default format for modern devices stays the same (VP9/WebM). [https://phabricator.wikimedia.org/T358266]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.15|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:46, 3 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30872536 --&gt;

== Tech News: 2026-33 ==

&lt;section begin=&quot;technews-2026-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/33|Translations]] are available.

'''Updates for editors'''
* [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] A new ChartWizard is [[c:Special:ChartWizard/Data:Example.Pie.chart|now available on Wikimedia Commons]] for users interested in creating charts from their own data. The wizard makes the [[mw:Special:MyLanguage/Extension:Chart|Chart extension]] more beginner-friendly by allowing editors to create charts, such as bar and pie charts, without needing to use JSON. Users can still switch to the JSON editor if they prefer. Feedback on the new tool is welcome on the [[m:Talk:Community Wishlist/W414|wish talk page]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the Wikipedia iOS app’s Picture of the Day widget displayed the same image every day instead of updating daily, has now been fixed. [https://phabricator.wikimedia.org/T430692]

'''Updates for technical contributors'''
* [[mw:Special:MyLanguage/Extension:Math|Math formula]] SVG images will soon be generated in the browser instead of on the server. MathML continues to be generated on the server and renders in the browser without JavaScript. Wikibooks will see this change on 12 August, Wikisource on 19 August and Wikipedia from 20-27 August. You can try this by selecting &quot;{{int:Mw-math-mathjax}}&quot; in your preferences. This change is part of [[mw:Special:MyLanguage/RESTBase/deprecation|deprecating RESTBase]] and [[phab:T431372|deprecating Mathoid]]. [https://phabricator.wikimedia.org/T271001]
* Category pages will soon support sorting entries by the time they are added to a category. This will make it easier to find recently or long-standing categorized pages. It will also improve workflows for maintenance categories such as deletion backlogs and other time-based review tasks. You can use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;cldsort=timestamp&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; URL argument in category view to sort the entries. [https://phabricator.wikimedia.org/T433768]
* [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] and user scripts on Wikimedia wikis may now use [[phab:T395347|ES2018 features]] and [[phab:T419142|ES2019 features]] in JavaScript code. Previously, the platform only allowed up to ES2017. MediaWiki validates the source code to protect functionality from syntax errors and to ensure scripts are valid in all [[mw:Special:MyLanguage/Compatibility#Browser_support_matrix|supported browsers]]. [https://phabricator.wikimedia.org/T419142]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.16|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:45, 10 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30901051 --&gt;

== Tech News: 2026-34 ==

&lt;section begin=&quot;technews-2026-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/34|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Worklist|Worklist feature]] for the Event Registration tool is now live on all Wikimedia wikis. With Worklist, event organizers can add the articles their event will focus on directly to the event page. The Worklist also powers [[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Worklist#How Event Pathways uses the worklist|Event Pathways]] which notifies other editors of the upcoming or ongoing event when they edit an article featured in the event's Worklist. This is the minimum viable version (MVP), and feedback is welcome. Organizers are encouraged to try the feature. A hands-on [[m:Special:MyLanguage/Event:Worklist Setup Workshop: Get Your Event Ready|Worklist Setup Workshop]] will take place on 18 August at 16:00 UTC and 19 August at 11:00 UTC.

'''Updates for editors'''
* [[Special:ShortPages]] displays short pages by their size, but in many cases it gets filled with disambiguations and soft redirects, making it harder to find the short articles themselves. Starting this weekend, you will be able to choose not to include an article in the special page by adding the magic word &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__EXPECTSHORTPAGE__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T433203]
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q3436680|Bole]] ([[w:bol:|&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;w:bol:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;]]) [https://phabricator.wikimedia.org/T429921]
* Starting the week of August 17, the page toolbar will wrap onto two lines when there is not enough horizontal space for all the buttons. This is a [[phab:T429518|fully merged patch from the Reader Experience team]] which aims to reduce crowding in the Vector 2022 toolbar, that may occur on some language Wikipedias at certain screen widths.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:16}} community-submitted {{PLURAL:16|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, uploading large files to Wikimedia Commons has become more stable and less prone to failure following some fixes related to the “Could not acquire lock” upload error. [https://phabricator.wikimedia.org/T386640]

'''Updates for technical contributors'''
* Debian Bullseye will reach the end of its Long Term Support on 31 August 2026. [[phab:T434103|Some Cloud VPS projects]] still have instances running Debian Bullseye. Maintainers of those projects are encouraged to migrate to Debian Bookworm or Debian Trixie. A [[wikitech:Help:Cloud VPS instance operating system migration|migration guide]] is available to help with the process, and users may also want to consider whether their workload is better suited to Toolforge. If you need help or cannot complete the migration by 31 August, please contact the Cloud VPS admins as soon as possible. [[listarchive:list/cloud-announce@lists.wikimedia.org/thread/RVIPQSYLKMSL5M46JP6NEJVGVE6I2RXQ/|Read more]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.16|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W34&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:03, 17 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30922806 --&gt;

== Tech News: 2026-35 ==

&lt;section begin=&quot;technews-2026-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/35|Translations]] are available.

'''Updates for editors'''
* The [[Special:CreateAccount|Special:CreateAccount]] page has been simplified as part of ongoing work to modernize the account creation experience. The panel showing project statistics no longer appears next to the form on desktop and mobile web. Multiple account creation experiments show that a simpler form helps newcomers complete registration. [https://phabricator.wikimedia.org/T433783]
* In order to improve page performance, images now load when they are viewed. This means images lower down an article will not load if a reader never scrolls to that part of the page, which may affect some image-related metrics. [https://phabricator.wikimedia.org/T148047]

* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:42}} community-submitted {{PLURAL:42|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where image thumbnails in Abstract Wikipedia could fail to display after the corresponding file was moved on Wikimedia Commons, has now been fixed. Thumbnails will now update correctly when files are moved. [https://phabricator.wikimedia.org/T433448]

'''Updates for technical contributors'''
* User Info card is a feature that helps patrollers see information about user accounts. So far, it has been available only in places such as page history, logs and recent changes. Now, it's possible to [[mw:Special:MyLanguage/Help:Extension:CheckUser#User_Info_card_in_page_content|place it in the page content]] as well, using the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{#uic:}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; parser function. It can be particularly useful in templates like [[:en:Template:Userlinks|&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{Userlinks}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;]] (or their specialized variants), as it will make it easier to see the context about a user on various noticeboard pages. The card will be displayed only to users who have it enabled in their [[Special:Preferences#mw-input-wpcheckuser-userinfocard-enable|preferences]]. [https://phabricator.wikimedia.org/T424466]
* Due to user security and privacy risks, we have disabled access to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Special:MyPage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; URLs when specifically using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=raw&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. If you are impacted by this, consider whether you can use an alternative approach. &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Special:MyPage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; URLs can still be accessed and used without &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=raw&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Specified user page URLs (e.g. &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;User:Myusername&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;) can still be used with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=raw&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T120386]
* Due to an update, the thumbnailing software has been improved. This includes upgrading &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;librsvg&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; to 2.60 and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;ImageMagick&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; to 7, as well as resolving a number of long-standing thumbnailing bugs like rendering errors. [https://phabricator.wikimedia.org/T419815#12222841]

* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.17|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:45, 24 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30961208 --&gt;

== Tech News: 2026-36 ==

&lt;section begin=&quot;technews-2026-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/36|Translations]] are available.

'''Weekly highlight'''
* A new format for the Community Wishlist is open for feedback. You can [[m:Special:MyLanguage/Community Wishlist/Community Wishlist 2027|read the proposed ideas on Meta]]. This new process plans to improve how wishes are triaged, voted on, and prioritized in a way that is transparent and balanced across project families and language editions. This consultation is open for two weeks.

'''Updates for editors'''
* The latest release of the Wikipedia Android app includes updates to the Saved feature, bringing the app’s saving experience closer to iOS and Web. The update redesigns the Saved tab with an “All articles” view, removes the default “Saved” reading list, renames reading lists to “Collections,” and modernizes the article-saving experience. [https://phabricator.wikimedia.org/T420788]
* The [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists]] feature was enabled for all logged-in users on Bengali, Chinese, Czech and Vietnamese Wikipedias on August 25, after several months as a beta feature. Reading Lists will be available to all logged-in users on Arabic, French and Indonesian Wikipedias on September 1, followed by English Wikipedia on September 14, and all other Wikipedia wikis on September 28.
* At the end of the month, some logged-out readers on Bengali, Czech, Persian, English, and Polish Wikipedias using the Minerva skin on mobile will see an [[mw:Special:MyLanguage/Readers/Reader_Growth/Minimal_Minerva|updated navigation bar]] in an [[w:A/B test|A/B test]]. The test will compare the current navigation bar with a new version designed to make it easier to find information more quickly. The goal is to determine whether these changes encourage readers to return more often. This experiment will not change the experience for logged-in readers and/or editors.
* Editors who maintain redirects, templates, and categories used on redirect pages now have improved ways for finding and curating redirects. Previously, redirects pages could not be searched. Two new search keywords, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;onlyredirects:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;withredirects:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, now allow redirects to be searched directly and can be combined with existing keywords such as &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;incategory:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;intitle:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;insource:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T204089]
* The ISBN lookup tools for generating citations were recently not working because of external service problems. Developers are working on solutions. [https://phabricator.wikimedia.org/T435179]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where searching for pages by category using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;deepcat&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; could return no results or unrelated results has now been fixed. [https://phabricator.wikimedia.org/T414859]

'''Updates for technical contributors'''
* The domain of URLs for thumbnails is changing from upload.wikimedia.org to thumb.wikimedia.org. The old URLs will continue to work for the foreseeable future but MediaWiki will advertise the new domain instead. URLs to other types of media such as original files, videos and transcodes will still be served from upload.wikimedia.org. [https://phabricator.wikimedia.org/T427465]
* The Wikimedia [https://www.mediawiki.org/w/index.php?api=wmf-math%2Fv1&amp;title=Special%3ARestSandbox Math API] is now deprecated. These endpoints will be fully sunset by the end of September 2026. Developers who call these endpoints should transition to alternative math rendering solutions, such as the native [https://developer.mozilla.org/en-US/docs/Web/MathML MathML] or [https://www.mathjax.org/ MathJax]. Third-party MediaWiki installations that utilize the Math extension for formula rendering are required to upgrade to v1.43+ to avoid disruption of service.

* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.18|MediaWiki]]

'''In depth'''
* Read more about [[mw:Special:MyLanguage/Edit_check/TextMatch|TextMatch]] in a Diff post titled, [[diffblog:2026/08/28/custom-edit-suggestions-for-every-wiki-how-communities-are-shaping-suggestion-mode-with-textmatch/|Custom edit suggestions for every wiki: How communities are shaping Suggestion Mode with TextMatch]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:53, 31 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30990659 --&gt;

== Tech News: 2026-37 ==

&lt;section begin=&quot;technews-2026-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/37|Translations]] are available.

'''Updates for editors'''
* [[mw:Special:MyLanguage/Help:Growth/Tools/Add_a_link|Add a Link]] has been upgraded for English Wikipedia users with improved detection of links that have different capitalization. This will help reduce ambiguous suggestions caused by differences in capitalization in article titles, including articles about cultural goods. A second phase of improvements is planned, which will also prepare the feature for release in other languages. [https://phabricator.wikimedia.org/T435526#12240876] [https://phabricator.wikimedia.org/T434259]
* The [[mw:Special:MyLanguage/Article_guidance|Article guidance]] feature will be enabled by default for junior editors on Simple English and Turkish Wikipedia starting 10 September 2026 following [[mw:Special:MyLanguage/Article_guidance/Updates#Summary_of_experiment_result|an experiment]]. Junior editors with 0 to 99 edits will automatically see the feature when they click a red link or use the &quot;[[tr:Vikipedi:Madde_sihirbazı|Madde oluşturto]]&quot; option on Turkish Wikipedia to create a new article. The change is intended to help junior editors create higher-quality articles that meet each Wikipedia’s standards. [[phab:maniphest/query/z3tcTjmLMxk5/#R|Additional improvements]] will continue based on the experiment results and community feedback.
* The [https://pageviews.wmcloud.org Pageviews Analysis] tool which allows users to compare pageviews across multiple pages, turns 10 years old this year and several new features have been added. They include [[toolforge:wikinav|WikiNav]] which provides insights into how readers of Wikipedia explore the content, editing stats in [https://pageviews.wmcloud.org/siteviews?range=latest-30&amp;sites=en.wikipedia.org Siteviews], the ability to [https://pageviews.wmcloud.org/massviews?source=wikiproject&amp;project=en.wikipedia.org lookup articles belonging to a WikiProject], and support for dark mode. [https://phabricator.wikimedia.org/T378549]
* A [[mw:Special:MyLanguage/Readers/Reader_Growth/Minimal_Minerva|visually simplified Minerva]] navigation bar is being tested for logged-out readers using mobile web on the Bengali, Czech, English, Farsi, and Polish Wikipedias. The experiment aims to determine whether simplifying the navigation improves reader retention. The test will run from August 31 to September 28, and no action is required from users.
* [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] is working on improving [[en:Wikipedia:VisualEditor/Named references|auto-generated reference names in VisualEditor]]. Editors will only notice a slight change starting this week. When adding automatic reference names the numbering will start at &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;:1&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; instead of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;:0&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Read more on the [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names|project page]]. [https://gerrit.wikimedia.org/r/c/VisualEditor/VisualEditor/+/1332704]
*  All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only for a few minutes]] on September 23. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T433363]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where [[mw:Special:MyLanguage/Parsoid|Parsoid]] could mishandle nested nowiki tags, causing content to be lost and displaying unwanted text has now been fixed. [https://phabricator.wikimedia.org/T435116]
'''Updates for technical contributors'''
* Interface administrators can configure gadgets from [[MediaWiki:Gadgets-definition]]. The definition format has been updated and no longer requires the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;ResourceLoader&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; option, as gadgets are always loaded through &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;ResourceLoader&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This simplifies gadget configuration by removing an option that is no longer necessary, making gadget definitions easier for administrators. [https://phabricator.wikimedia.org/T298199]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.19|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:44, 7 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31008437 --&gt;

== Tech News: 2026-38 ==
&lt;section begin=&quot;technews-2026-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/38|Translations]] are available.

'''Weekly highlight'''
* The Future Audiences team has started a [[en:Special:MyLanguage/Wikipedia:Village pump (proposals)#New proposed experiment to let readers know about “Google preferred sources”|discussion on English Wikipedia]] about a proposed new experiment to show a temporary notice about the new [https://blog.google/products-and-platforms/products/search/original-high-quality-content-search/ Google Preferred Sources] feature to Wikipedia readers coming from Google. The experiment is to test whether this notice makes visitors come back to Wikipedia more often. Preferred Sources feature lets users choose websites they trust so Google can highlight content from those sources more prominently in Search and AI experiences. We are interested in testing this on other languages that Google supports and would welcome assistance with starting conversations on other wikis. If interested, please reach out to Future Audiences on the [[m:Special:MyLanguage/Future Audiences/Preferred sources|talk page]].

'''Updates for editors'''
* The chat platform Discord is releasing a new self-service framework that websites can use to specify how their links should look when shared on Discord. The Future Audiences team is planning to start using this new framework to improve how Wikipedia links look when shared on Discord, giving better attribution and credit to contributors. The link appearance and behavior won't change in the first phase of this project as we want to first collect some baseline data to assess the impact of future changes. However, if community members who are active on Discord spot any issues, they can reach out on [[phab:tag/future-audiences/|Phabricator]] or [[m:Special:MyLanguage/Future Audiences|Metawiki]].
* The Reader Growth team will be running an experiment to test whether [[mw:Special:MyLanguage/Readers/Reader_Growth/Compact Lead|compacting article lead sections on mobiles]] with the addition of a &quot;read more&quot; button improves reader retention. The test will begin on September 17 on Arabic, Spanish, French, Indonesian, Italian, Japanese, Portuguese, Vietnamese, and Chinese Wikipedias and will run for four weeks.
* All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only for a few minutes]] on September 23. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming week. [https://phabricator.wikimedia.org/T433363]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where [https://wikistats.wmcloud.org/ Wikistats] for Wikipedias was returning an HTTP 500 error and could not be reached, has now been fixed. [https://phabricator.wikimedia.org/T435959]

'''Updates for technical contributors'''
* Developers who maintain a tool that queries the Wikimedia Commons links tables need to update their code to connect to the new x4 database cluster. The links tables have been moved from the s4 cluster to x4, and will no longer receive updates on s4. The page and redirect tables remain available on both clusters. A wiki replica for the x4 cluster will be set up afterwards. The change is being made because the s4 cluster has grown too large to operate efficiently. You can [[wikitech:Special:MyLanguage/News/2026 Commons links tables database split|read more]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.20|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:31, 14 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31039826 --&gt;

== Tech News: 2026-39 ==
&lt;section begin=&quot;technews-2026-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/39|Translations]] are available.

'''Weekly highlight'''
* [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 23 September 2026 at [https://zonestamp.toolforge.org/1790172000 14:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Special:MyLanguage/Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies. [https://phabricator.wikimedia.org/T433363]

'''Updates for editors'''
* The Growth team tested a new post-edit notice designed to [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#4. Encourage Temporary Accounts to Register|encourage Temporary Account holders to register for a permanent account]]. The new notice replaced multiple dialogs and a simultaneous welcome notification with a single message highlighting the benefits of creating an account. The experiment increased permanent account creation from 1.94% to 3.66%, an 87% relative increase. The change will now be released to all wikis. [https://phabricator.wikimedia.org/T433551]
* For Wikipedia editors using the [[Special:Preferences#mw-prefsection-betafeatures|&quot;Suggestion mode&quot; Beta Feature]], there is a [[Special:Preferences#mw-input-wpvisualeditor-editcheck-experimental|new opt-in user preference]] for showing &quot;experimental&quot; edit checks and suggestions. These types are listed at [[Special:EditChecks#experimental-checks|Special:EditChecks]], and are intended for early developer-testing and for gathering feedback from experienced users. Administrators can change the [[mw:Special:MyLanguage/Edit check/Configuration|configuration details]] for each suggestion as usual. Administrators can also [[mw:Special:MyLanguage/Help:Suggestion mode#Experimental Suggestions|use this feature]] to help test their community's ideas for locally created checks and suggestions. Experimental types will have a more distinct visual style later this week. [[mw:Talk:VisualEditor/Suggestion Mode|Feedback is welcome]].
* The [[m:CEE Technical Village Pump|CEE Technical Village Pump]] has been launched as a space for technical discussions and collaboration among Wikimedia communities in Central and Eastern Europe.
* A new tool, [[mw:Special:MyLanguage/Language Onboarding and Development/Starter kit|Starter Kit]], is now available for new and small language Wikipedia communities. It brings together guided tasks, tools, and resources to help communities get started, monitor their progress, and collaborate with the broader Wikimedia community. Starter Kit is hosted on Wikimedia Toolforge and is designed for Wikipedias with fewer than 50,000 articles. Learn more about how Starter Kit works and how communities have been using it in [[diffblog:2026/09/18/introducing-starter-kit-for-new-and-small-language-wikipedias/|this Diff blog post]].
* The Reader Growth team is launching a retest of the [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|image carousel experiment]]. The retest will use three new versions of the design, updated based on community feedback. The team will assess results and determine with communities whether or not to proceed with the feature. The experiment will begin the week of September 28 and will run on Arabic, Bengali, Chinese, Czech, English, Farsi, French, German, Indonesian, Japanese, Polish, Portuguese, Spanish, Swedish, and Vietnamese Wikipedias.
* After successful rollouts to Arabic, Bengali, Chinese, Czech, English, French, Indonesian, and Vietnamese Wikipedias, the [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists feature]] from the Reader Experience team will be available to all logged-in users on all Wikipedia wikis starting September 28.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the mentor status script repeatedly updated mentors’ status even when nothing had changed, creating unnecessary Recent Changes entries, has now been fixed. [https://phabricator.wikimedia.org/T436659]

'''Updates for technical contributors'''
* The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;abusefilters&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; list query API, used to retrieve specific details about active or historical abuse filters configured on a wiki, has been updated to support the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;formatversion=2&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; configuration parameter, thereby leading to breaking changes to the API response. All users who maintain user-scripts or other code need to check if they use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;abusefilters&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; list query API, and respond to any breaking changes. [https://phabricator.wikimedia.org/T435828]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.21|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 11:56, 27 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31085309 --&gt;

== Tech News: 2026-40 ==
&lt;section begin=&quot;technews-2026-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/40|Translations]] are available.

'''Updates for editors'''
* Users of default skins on desktop and mobile web (Vector 2022 and Minerva respectively) will now be informed if the search suggestions displayed as they type into Search come from page redirects. Previously, users could get confused when the top search suggestions did not match exactly what they had typed. The new redirect notice clarifies that these suggestions are not random, but related to the search. The suggested pages may be redirects to the most relevant destination articles being searched for. [https://phabricator.wikimedia.org/T303013]
* Logged-out Wikipedia users on mobile web will begin to see a bookmark button for saving articles to their [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|reading lists]] instead of the current Watchstar button. When clicked, the bookmark button prompts users to log in or create an account to save an article for later. An earlier experiment found that the bookmark icon was four times more likely than the Watchstar button to encourage logged-out users to create an account. This change is part of efforts to encourage more readers to become account holders and use features that help them save and return to content. [https://phabricator.wikimedia.org/T438769]
* An experiment will begin on September 29 on Spanish, Arabic, English, and French Wikipedias to simplify the experience users encounter immediately after account creation. The test will explore whether removing friction from the current Welcome Survey helps new users understand what to do next more easily. The experiment will run through the end of October. [https://phabricator.wikimedia.org/T430058]

* The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] has put out a call for the community to suggest topics that it should consider and discuss with the Wikimedia Foundation. Editors and technical contributors are invited to add topics to [[m:Talk:Product and Technology Advisory Council|the council's talk page]].
* Later this week, it will be possible for communities to configure [[mw:Special:MyLanguage/Help:Edit check|Edit Checks and Suggestions]] so that they are [[mw:Special:MyLanguage/Edit check/Configuration#extraNamespaces|shown within additional namespaces]], such as a &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Draft:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; namespace. [[mw:Talk:VisualEditor/Suggestion Mode|Feedback is welcome]].
* A bug in the [[m:Special:GlobalWatchlist|Global Watchlist]] is causing the page to fail to load for users that have unseen changes on Wikimedia Commons. Developers are working on a fix. Until then, affected users can follow the workarounds described in the [[mw:Special:MyLanguage/Extension:GlobalWatchlist#GlobalWatchlist temporarily broken for users with Wikimedia Commons|relevant]] help section.
* New magic words &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{CATEGORYSORT:TIMESTAMP}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{CATEGORYSORT:RTIMESTAMP}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; are now available for use. They are able to change the default sorting of categories to be based on timestamp of categorization. [https://phabricator.wikimedia.org/T433768]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].  For example, the issue where [https://wikistats.wmcloud.org/ Wikistats] for Wikipedias was not reachable has now been fixed. [https://phabricator.wikimedia.org/T435959]

'''Updates for technical contributors'''
* The datacenter switchover, scheduled on September 23, has been postponed. [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|The equinox exercise]] revealed capacity issues, preventing the switch of all services to the other datacenter. The process has been paused, prioritizing investigating that issue, to ensure that we continue to be able to serve our users reliably. A new exercise will be scheduled. Meanwhile, all traffic and edits continue to work as usual.
* On September 23, a power loss briefly affected all wikis, creating intermittent issues on both reading and editing modes. This also affected Gerrit. This is unrelated to the datacenter switchover postponement. [https://www.wikimediastatus.net/incidents/9fm19t8qykk8]
* [[mw:Special:MyLanguage/Help:Extension:Produnto|Produnto]] has been deployed to mediawiki.org and [[phab:T421436|some Indic language wikis]]. We want to hear your feedback on this experimental product. Produnto allows users to use Lua modules hosted in [[gitlab:repos/lua|Wikimedia's GitLab]]. It simplifies the sharing of Lua code between wikis. Instead of copying each individual Lua module from one wiki to another, users will be able to easily share packages across all wikis.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.22|MediaWiki]]

'''Meetings and events'''
* A [[m:Event:Cross-wiki Code Collaboration Workshop|Cross-wiki code collaboration workshop]] will be held online on October 2 at 12:30 UTC, bringing together South Asian technical contributors to learn about Produnto and begin piloting it on Hindi, Punjabi, Odia, Telugu, and Malayalam Wikipedias. Produnto is a package manager for deploying Lua modules hosted on GitLab to Wikimedia wikis, and was recently deployed on some pilot wikis.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 10:49, 28 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31092225 --&gt;</text>
      <sha1>bu8q2bhifx51w5rbdhfu3nk99xzc8wj</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:WikiJournal of Science/Editors</title>
    <ns>1</ns>
    <id>206506</id>
    <revision>
      <id>2834785</id>
      <parentid>2790208</parentid>
      <timestamp>2026-09-28T03:53:08Z</timestamp>
      <contributor>
        <username>~2026-52029-76</username>
        <id>3111737</id>
      </contributor>
      <origin>2834785</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12984" sha1="q57xeal8l9ijgadgr4nfaxoacve6ppu" xml:space="preserve">&lt;noinclude&gt;
{{WikiJournal editorial application top
|archive box = {{Archive box|[[/Archive 2016]]&lt;br&gt;[[/Archive 2017]]&lt;br&gt;[[/Archive 2018]]&lt;br&gt;[[/Archive 2019]]&lt;br&gt;[[/Archive 2020]]&lt;br&gt;[[/Archive 2022]]&lt;br&gt;[[/Archive 2023]]
}}
}}
&lt;/noinclude&gt;

==Associate editor application of OGUNJIMI OLUWOLE TEMIDAYO==

{{WikiJournal editor application submitted
 | position                =Associate editor
 | name                    =OGUNJIMI OLUWOLE TEMIDAYO
 | qualifications          =M.Sc. Geography and Planning
 | link                    =Linkedin.com/in/oluwole-ogunjimi
 | areas_of_expertise      =Enivironmental Management and Data Analysis
 | professional_experience =An administrator with over five years of working experience in both healthcare and education sectors. I have a Master’s degree from the University of Lagos and a Bachelor's degree from the University of Ibadan, I have developed significant research skills, especially in analyzing the impacts of human activities on water quality and understanding the socio-economic effects of environmental changes. With hands-on experience in geospatial mapping, data analytics, and programming (Python), he combines his passion for environmental conservation with a commitment to translating research findings into actionable insights.
 | publishing_experience   =Environmental and Socio-Economic Impact of Artisanal Mining in Oke Ogun Region, 2016. https://drive.google.com/file/d/1l9WTh-ZhUpKbRN_TP28oP-mvlaiWGf0S/view?usp=drive_link
Impact of Human Activities on Water Quality of Kudeti River, Ibadan, Oyo State, Nigeria, 2016.
https://drive.google.com/file/d/1N00Eo1RitC5dKoX38G1H8r7MJsKdD3Aw/view?usp=drive_link
Impact of Rural Transportation on The Distribution of Agricultural Produce in Obafemi Owode Local Government Area, Ogun State, Nigeria, 2023.
https://drive.google.com/file/d/1YMYpcdC1JqnxQO-fq5eM3eitbHpQH-HB/view?usp=drive_link
 | open_experience         =I am an active Wikipedia editor with over 300 edits.  Significant editing history on English Wikipedia under username &quot;Kamoranesi90&quot;.
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 12:35, 18 October 2024 (UTC)
}}
* {{re|Kamoranesi90}} Please adjust your security settings for the 3 Google Drive links. I am unable to view them. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:32, 25 October 2024 (UTC)
*:Thank you @[[User:OhanaUnited|OhanaUnited]]. I have adjusted the security settings.  [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 14:49, 25 October 2024 (UTC)
*::Thank you. Do you have any publication examples which are peer-reviewed? For example, these could be journal articles or book chapters. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 20:12, 25 October 2024 (UTC)
*:::No, I don't but I am working on publishing my research works. Thank you [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 09:19, 26 October 2024 (UTC)
*::::I am open to '''support''' your associate editor application with mentorship from board members to support your growth in gaining experience around the publishing area. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 18:52, 14 November 2024 (UTC)
*:::::Wow. That's great. I am open to learning from your wealth of knowledge and experience. I appreciate your kind gesture. 
*:::::Thank you. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 18:59, 14 November 2024 (UTC)
*::::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 21:44, 18 November 2024 (UTC)
*:::::::Thank you. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 02:59, 22 November 2024 (UTC)
*If [[User:OhanaUnited]] is happy to mentor, I think you would be a valuable addition to the team, so I would be willing to support. Associate editor positions don't need to be as experienced as board members, just willing to learn and able to work well with others, especially if someone is able to take the lead if calleging scenarios arise (e.g. confilcting peer reviews). [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 02:48, 9 January 2025 (UTC)
*'''Support application''' Though I lack experience in such matters, I have recently learned enough to recognize this to be an easy decision. Regarding mentoring, I think I could use some mentoring myself.  Whenever appropriate, please CC me on mentor-mentee communications.  In particular, I am having trouble dealing with the large number of referee invites required to recruit volunteers. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:50, 11 January 2025 (UTC)

{{re|Kamoranesi90}} My apologies for the delay in getting back to you. 

'''Result: Accepted into the editorial board as associate editor.''' 

: [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add &lt;code&gt;DONE&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{Done}}&lt;/nowiki&gt;&lt;/code&gt; after someone has performed the task):

# [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) 
{{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}}
# Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template
# Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata
# Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&amp;utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members
# Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed
# Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]]
[[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 20:12, 2 April 2025 (UTC)

{{ping|Kamoranesi90}}, can you confirm if you're still interested in joining? I have emailed you through the wiki's email function a while back but did not receive a response from you. I also noticed that [https://guc.toolforge.org/?by=date&amp;user=Kamoranesi90 you haven't edited in any WMF projects since November 2024] so I don't know if you still monitor your account for notifications. If I don't hear from you by the end of this month, I will archive this application without prejudice (i.e. you are more than welcome to continue the process to join as an associate editor when you replied back to this message). [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 03:43, 2 January 2026 (UTC)

==Editorial board application of Max Loomes==

{{WikiJournal editor application submitted
 | position                =Editorial board
 | name                    =Max Loomes
 | qualifications          =Bachelor of psychology (Hons I)(USYD), Master of Clinical Psychology (UTS), PhD Candidate (UTS)
 | link                    =https://profiles.uts.edu.au/student_Max.W.Loomes
 | areas_of_expertise      =Mental health, psychiatry, public health
 | professional_experience =Research (9 years, ~4 institutions, various areas both qualitative and quantitative, field work [ecology/education psychology], preclinical trials, project managing, clinical practice (trauma, schizophrenia), teaching (~1 year at 2 institutions for psychology undergraduate and post-graduate, some masters).
 | publishing_experience   =Published ~12 articles/chapters, reviewed for three different journals
 | open_experience         =Have updated various wikipedia pages and big believer in open access and dismantling the highly capitalistic structures that govern journal publishing
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Maxwloomes|Maxwloomes]] ([[User talk:Maxwloomes|discuss]] • [[Special:Contributions/Maxwloomes|contribs]]) 06:34, 13 May 2025 (UTC)
}}

:&lt;nowiki&gt;Great to see this application! I support it; have known Max for several years and he is passionate about many of the values behind WikiJournal  ~~~&lt;/nowiki&gt; [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 20:46, 4 February 2026 (UTC)


--[[User:Maxwloomes|Maxwloomes]] ([[User talk:Maxwloomes|discuss]] • [[Special:Contributions/Maxwloomes|contribs]]) 06:34, 13 May 2025 (UTC)

:* '''Support'''. Applicant has experience in research, publishing and open access (both Wikipedia and journal). [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 00:23, 27 May 2025 (UTC)

==Editorial board application of Alex O. Holcombe==

{{WikiJournal editor application submitted
 | position                =Editorial board
 | name                    =Alex O. Holcombe
 | qualifications          =PhD in psychology
 | link                    =https://profiles.sydney.edu.au/alex.holcombe
 | areas_of_expertise      =Psychology, neuroscience
 | professional_experience =Full-time academic position for 20 years
 | publishing_experience   =Advisory board or editorial board member of journals and initiatives including PLOS ONE, Collabra:Psychology, Replication Research, Meta-ROR, Meta-psychology...
 | open_experience         =Have been an Associate Editor of WikiJournalOfScience for the last couple years, handled three articles to publication, and have attended most, I think, of the monthly meetings in the last two years
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 20:56, 4 February 2026 (UTC)
}}
: I '''strongly support''' promoting Alex to be a full editorial board member. He has the handling editor for a number of submissions, e.g. [[WikiJournal of Science/Hypericum sechmenii|1]], [[WikiJournal of Science/The Himalayan fossil hoax|2]] and [[WikiJournal of Science/Popular Pet Reptile, the Leopard Gecko (Eublepharis macularius), Spontaneously Uses Running Wheel– Is It Locomotion Play?|3]]). He also gave insightful comments towards the operations of the WikiJournal of Science editorial board and the broader WikiJournal publishing umbrella. We need the expertise like Alex to further strengthen the board's expertise. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 18:03, 5 February 2026 (UTC)

==Associate editor application of Neil Vedwan==

{{WikiJournal editor application submitted
 | position                =Associate editor
 | name                    =Neil Vedwan
 | qualifications          =
 | link                    =
 | areas_of_expertise      =Type theory, Formal Methods, Computational Complexity, Computability, Category Theory, Set Theory
 | professional_experience =Researcher @ Montclair State university
 | publishing_experience   =Reviewer at zbMath Open, In the process of getting 2 papers published
 | open_experience         =Contributor to CSLib (a library of Lean 4), Wikipedia contributor
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[Special:Contributions/~2026-52029-76|~2026-52029-76]] ([[User talk:~2026-52029-76|talk]]) 03:53, 28 September 2026 (UTC)
}}</text>
      <sha1>q57xeal8l9ijgadgr4nfaxoacve6ppu</sha1>
    </revision>
    <revision>
      <id>2834816</id>
      <parentid>2834785</parentid>
      <timestamp>2026-09-28T06:52:35Z</timestamp>
      <contributor>
        <username>OhanaUnited</username>
        <id>18921</id>
      </contributor>
      <comment>/* Associate editor application of Neil Vedwan */ not logged in</comment>
      <origin>2834816</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13405" sha1="8izu6rekc3w3pukt6oa00tb5zp1kqnw" xml:space="preserve">&lt;noinclude&gt;
{{WikiJournal editorial application top
|archive box = {{Archive box|[[/Archive 2016]]&lt;br&gt;[[/Archive 2017]]&lt;br&gt;[[/Archive 2018]]&lt;br&gt;[[/Archive 2019]]&lt;br&gt;[[/Archive 2020]]&lt;br&gt;[[/Archive 2022]]&lt;br&gt;[[/Archive 2023]]
}}
}}
&lt;/noinclude&gt;

==Associate editor application of OGUNJIMI OLUWOLE TEMIDAYO==

{{WikiJournal editor application submitted
 | position                =Associate editor
 | name                    =OGUNJIMI OLUWOLE TEMIDAYO
 | qualifications          =M.Sc. Geography and Planning
 | link                    =Linkedin.com/in/oluwole-ogunjimi
 | areas_of_expertise      =Enivironmental Management and Data Analysis
 | professional_experience =An administrator with over five years of working experience in both healthcare and education sectors. I have a Master’s degree from the University of Lagos and a Bachelor's degree from the University of Ibadan, I have developed significant research skills, especially in analyzing the impacts of human activities on water quality and understanding the socio-economic effects of environmental changes. With hands-on experience in geospatial mapping, data analytics, and programming (Python), he combines his passion for environmental conservation with a commitment to translating research findings into actionable insights.
 | publishing_experience   =Environmental and Socio-Economic Impact of Artisanal Mining in Oke Ogun Region, 2016. https://drive.google.com/file/d/1l9WTh-ZhUpKbRN_TP28oP-mvlaiWGf0S/view?usp=drive_link
Impact of Human Activities on Water Quality of Kudeti River, Ibadan, Oyo State, Nigeria, 2016.
https://drive.google.com/file/d/1N00Eo1RitC5dKoX38G1H8r7MJsKdD3Aw/view?usp=drive_link
Impact of Rural Transportation on The Distribution of Agricultural Produce in Obafemi Owode Local Government Area, Ogun State, Nigeria, 2023.
https://drive.google.com/file/d/1YMYpcdC1JqnxQO-fq5eM3eitbHpQH-HB/view?usp=drive_link
 | open_experience         =I am an active Wikipedia editor with over 300 edits.  Significant editing history on English Wikipedia under username &quot;Kamoranesi90&quot;.
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 12:35, 18 October 2024 (UTC)
}}
* {{re|Kamoranesi90}} Please adjust your security settings for the 3 Google Drive links. I am unable to view them. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:32, 25 October 2024 (UTC)
*:Thank you @[[User:OhanaUnited|OhanaUnited]]. I have adjusted the security settings.  [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 14:49, 25 October 2024 (UTC)
*::Thank you. Do you have any publication examples which are peer-reviewed? For example, these could be journal articles or book chapters. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 20:12, 25 October 2024 (UTC)
*:::No, I don't but I am working on publishing my research works. Thank you [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 09:19, 26 October 2024 (UTC)
*::::I am open to '''support''' your associate editor application with mentorship from board members to support your growth in gaining experience around the publishing area. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 18:52, 14 November 2024 (UTC)
*:::::Wow. That's great. I am open to learning from your wealth of knowledge and experience. I appreciate your kind gesture. 
*:::::Thank you. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 18:59, 14 November 2024 (UTC)
*::::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 21:44, 18 November 2024 (UTC)
*:::::::Thank you. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 02:59, 22 November 2024 (UTC)
*If [[User:OhanaUnited]] is happy to mentor, I think you would be a valuable addition to the team, so I would be willing to support. Associate editor positions don't need to be as experienced as board members, just willing to learn and able to work well with others, especially if someone is able to take the lead if calleging scenarios arise (e.g. confilcting peer reviews). [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 02:48, 9 January 2025 (UTC)
*'''Support application''' Though I lack experience in such matters, I have recently learned enough to recognize this to be an easy decision. Regarding mentoring, I think I could use some mentoring myself.  Whenever appropriate, please CC me on mentor-mentee communications.  In particular, I am having trouble dealing with the large number of referee invites required to recruit volunteers. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:50, 11 January 2025 (UTC)

{{re|Kamoranesi90}} My apologies for the delay in getting back to you. 

'''Result: Accepted into the editorial board as associate editor.''' 

: [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add &lt;code&gt;DONE&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{Done}}&lt;/nowiki&gt;&lt;/code&gt; after someone has performed the task):

# [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) 
{{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}}
# Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template
# Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata
# Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&amp;utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members
# Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed
# Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]]
[[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 20:12, 2 April 2025 (UTC)

{{ping|Kamoranesi90}}, can you confirm if you're still interested in joining? I have emailed you through the wiki's email function a while back but did not receive a response from you. I also noticed that [https://guc.toolforge.org/?by=date&amp;user=Kamoranesi90 you haven't edited in any WMF projects since November 2024] so I don't know if you still monitor your account for notifications. If I don't hear from you by the end of this month, I will archive this application without prejudice (i.e. you are more than welcome to continue the process to join as an associate editor when you replied back to this message). [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 03:43, 2 January 2026 (UTC)

==Editorial board application of Max Loomes==

{{WikiJournal editor application submitted
 | position                =Editorial board
 | name                    =Max Loomes
 | qualifications          =Bachelor of psychology (Hons I)(USYD), Master of Clinical Psychology (UTS), PhD Candidate (UTS)
 | link                    =https://profiles.uts.edu.au/student_Max.W.Loomes
 | areas_of_expertise      =Mental health, psychiatry, public health
 | professional_experience =Research (9 years, ~4 institutions, various areas both qualitative and quantitative, field work [ecology/education psychology], preclinical trials, project managing, clinical practice (trauma, schizophrenia), teaching (~1 year at 2 institutions for psychology undergraduate and post-graduate, some masters).
 | publishing_experience   =Published ~12 articles/chapters, reviewed for three different journals
 | open_experience         =Have updated various wikipedia pages and big believer in open access and dismantling the highly capitalistic structures that govern journal publishing
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Maxwloomes|Maxwloomes]] ([[User talk:Maxwloomes|discuss]] • [[Special:Contributions/Maxwloomes|contribs]]) 06:34, 13 May 2025 (UTC)
}}

:&lt;nowiki&gt;Great to see this application! I support it; have known Max for several years and he is passionate about many of the values behind WikiJournal  ~~~&lt;/nowiki&gt; [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 20:46, 4 February 2026 (UTC)


--[[User:Maxwloomes|Maxwloomes]] ([[User talk:Maxwloomes|discuss]] • [[Special:Contributions/Maxwloomes|contribs]]) 06:34, 13 May 2025 (UTC)

:* '''Support'''. Applicant has experience in research, publishing and open access (both Wikipedia and journal). [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 00:23, 27 May 2025 (UTC)

==Editorial board application of Alex O. Holcombe==

{{WikiJournal editor application submitted
 | position                =Editorial board
 | name                    =Alex O. Holcombe
 | qualifications          =PhD in psychology
 | link                    =https://profiles.sydney.edu.au/alex.holcombe
 | areas_of_expertise      =Psychology, neuroscience
 | professional_experience =Full-time academic position for 20 years
 | publishing_experience   =Advisory board or editorial board member of journals and initiatives including PLOS ONE, Collabra:Psychology, Replication Research, Meta-ROR, Meta-psychology...
 | open_experience         =Have been an Associate Editor of WikiJournalOfScience for the last couple years, handled three articles to publication, and have attended most, I think, of the monthly meetings in the last two years
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 20:56, 4 February 2026 (UTC)
}}
: I '''strongly support''' promoting Alex to be a full editorial board member. He has the handling editor for a number of submissions, e.g. [[WikiJournal of Science/Hypericum sechmenii|1]], [[WikiJournal of Science/The Himalayan fossil hoax|2]] and [[WikiJournal of Science/Popular Pet Reptile, the Leopard Gecko (Eublepharis macularius), Spontaneously Uses Running Wheel– Is It Locomotion Play?|3]]). He also gave insightful comments towards the operations of the WikiJournal of Science editorial board and the broader WikiJournal publishing umbrella. We need the expertise like Alex to further strengthen the board's expertise. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 18:03, 5 February 2026 (UTC)

==Associate editor application of Neil Vedwan==

{{WikiJournal editor application submitted
 | position                =Associate editor
 | name                    =Neil Vedwan
 | qualifications          =
 | link                    =
 | areas_of_expertise      =Type theory, Formal Methods, Computational Complexity, Computability, Category Theory, Set Theory
 | professional_experience =Researcher @ Montclair State university
 | publishing_experience   =Reviewer at zbMath Open, In the process of getting 2 papers published
 | open_experience         =Contributor to CSLib (a library of Lean 4), Wikipedia contributor
 | policy_confirm          =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[Special:Contributions/~2026-52029-76|~2026-52029-76]] ([[User talk:~2026-52029-76|talk]]) 03:53, 28 September 2026 (UTC)
}}

:It looks like you're not logged in. Please log in and attach your signature to this application. Furthermore, I am unable to find a research by this name in Montclair State University's website with your area of expertise. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 06:52, 28 September 2026 (UTC)</text>
      <sha1>8izu6rekc3w3pukt6oa00tb5zp1kqnw</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Assessment/Topic</title>
    <ns>0</ns>
    <id>221601</id>
    <revision>
      <id>2834728</id>
      <parentid>2830324</parentid>
      <timestamp>2026-09-27T22:14:38Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Examples */ + 2026</comment>
      <origin>2834728</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22935" sha1="8jtb6a10pvx1yeghekpr2amfqfuhui5" xml:space="preserve">{{title|Topic development — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Develop a chapter plan and user page''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
{{countdown
 |year       = 2025
 |month      = 08
 |day        = 14
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}}
--&gt; &lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester --&gt;{{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt;
{{TOCright}}
==Overview==
* Weight: 10%
* Due {{/Due}}
* Tasks
** Create a Wikiversity user account.
** [[Motivation and emotion/Assessment/Selection|Select or negotiate]] an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]].
** Build wiki editing skills.
** Develop a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] on Wikiversity which consists of:
*** Title and sub-title
*** Headings (and possibly sub-headings)
*** Key points for each section (and sub-section)
*** Figure (at least 1)
*** Learning feature (plan at least 1)
*** References (6+ relevant, high quality sources)
*** Resources (2+ see also and 2+ external links)
** Create a Wikiversity user page:
*** Introduce yourself
*** Summarise at least three different types of social contributions on your Wikiversity user page
* Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].
* Guidance for this assignment is provided in Module 1:
** [[Motivation and emotion/Lectures/Introduction|Lecture 01]]
** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]]
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]

==Purpose==
The primary purpose of the topic development is to share your ideas and plans to date, enabling feedback that can help refine and strengthen the book chapter. The secondary purpose is to encourage development of collaborative wiki editing skills.

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing marking rubric].
*Marks will be provided via {{Motivation and emotion/Canvas}} by Census Date (end of Week 4).
* Written feedback to help guide [[Motivation and emotion/Assessment/Chapter|book chapter]] drafting will be provided via the topic's Wikiversity discussion page.
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions.

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence.
* Submissions will be accepted up to 3 days late (-10% per day late).
* If you don't submit this assessment, withdrawal from the unit by Census Date (end of Week 4) is recommended.

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|- style=&quot;vertical-align:top;&quot;
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|- style=&quot;vertical-align:top;&quot;
| Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic.
|- style=&quot;vertical-align:top;&quot;
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate your ideas by sharing a chapter plan; provide feedback on other plans.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic and submitting an on-time chapter plan.
|-
| style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Learn how to edit in a collaborative, online environment.
|}

==Instructions==
The topic development should communicate your current thinking and plans for the project. It is not expected to be a fully developed or final product. Outline your ideas as they currently stand, even if these are still evolving. Focus on clearly communicating the proposed direction so that constructive feedback can be provided.

The following instructions should be used to guide the topic development:
* Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of:
*# Title and sub-title (pre-approved or [[Motivation and emotion/Assessment/Selection#New topics|negotiated]])
*# Overview
*# 3 to 5 other top-level headings (subheadings are optional)
*#* Key points for each heading and subheading, with citations
*#* 1+ relevant figure(s)
*#* 1+ actual or planned learning feature
*# Conclusion
*# See also
*#* 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article)
*# References (at least 6, which are cited)
*# External links
*#* 2+ external links (to external resources)
*# Wikiversity user page
*#* Self-introduction
*#* A link to the chapter being worked on
*#* Social contributions in a numbered list with a summary and direct link to evidence:
*#** 1 direct edit to improve another book chapter (past or present)
*#** 1 talk page comment on another book chapter (past or present)
*#** 1 {{Motivation and emotion/Canvas}} discussion post
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* &lt;span id=&quot;Word count&quot;&gt;Length (Word count):&lt;/span&gt; There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700).
* Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments

==Template==
{{:Motivation and emotion/Assessment/Topic/Quickstarttip}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]

Topic developments will be marked against the following criteria.

{{anchor|Title}}
===Title and sub-title (10%)===
* Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}})
* Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]]
* Do not include user name; authorship is as per the page's editing history

{{anchor|Headings}}

===Headings (10%)===
* Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links)
* Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading.
* The top-level headings should align with the sub-title and focus questions
* Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]])

{{anchor|Overview}}

===Overview (10%)===
* A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
* At least 3 bullet points outlining the &quot;problem&quot; (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]]
* 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]

{{anchor|Key points}}

===Key points (10%)===
* At least 3 bullet points per section (i.e., per heading or sub-heading)
* Overview the most relevant theory(ies), including key citations
* Overview the most relevant research, including key citations
* Provide at least 1 introductory bullet point before branching into sub-sections
* Address the problem (i.e., answer the question in the sub-title)

{{Anchor|Figure}}
===Figure (10%)===
* Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]].
* Number each figure sequentially (e.g., Figure 1, Figure 2 etc.)
* Include a descriptive caption that connects the figure to the text
* Cite each figure at least once in the main text (e.g., see Figure 1)
* Optimise image display size to make it easy to read (i.e., not too big or too small)

{{Anchor|Learning feature}}
===Learning feature (10%)===
* In addition to the scenario in the Overview, include at least 1 of the following learning features:
** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
** Internal (wiki) links:
*** At least 1 embedded link (i.e., in main body text) to a relevant book chapter, and
*** At least 1 embedded link (i.e., in main body text) to a relevant Wikipedia article
** Quiz question with correct and incorrect answers
** Table with an APA style caption

{{anchor|References}}

===References (10%)===
* Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]])
* Each source should be cited at least once in the key points
* Include a balance of key theoretical and key research articles

{{anchor|Resources}}

===Resources (10%)===
* '''See also''' (Level 2 heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article)
** Provide at least 1 bullet-pointed:
*** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter
*** internal wiki link to a relevant Wikipedia page
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., Book chapter, 2023)
** Use alphabetical order
* '''External links''' (Level 2 heading): Provide at least 2 external links to key internet resources
** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles)
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., The Conversation)
** Use alphabetical order

{{anchor|User page}}

===User page (10%)===
* Create a Wikiversity user page for your user account
* Edit the user page to provide information about yourself
* Recommended headings:
** About me
** Book chapter I'm working on
*** Include an internal (wiki) link to the chapter page
** Social contributions
* Consider linking to your other online profiles

{{anchor|Social contribution}}
{{anchor|Socialcontribution}}

===Social contribution (10%)===
* On your Wikiversity user page, summarise and link to ''direct evidence'' that you made at least 3 different types of contributions:
*# direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics)
*# provided feedback by commenting on a book chapter's talk page (current or previous topics)
*# posted about motivation or emotion or the assessment tasks to the {{Motivation and emotion/Canvas}} discussion forum&lt;!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --&gt;
* [[Motivation and emotion/Wikiversity/Social contributions|More info]]

== Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]}}

==Examples==
;About
* Below are some examples of topic development submissions which received 100%
* The links go to snapshots of topic pages and user pages as submitted for the topic development; these are not the final book chapter submissions
* It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter)

;2026
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Adaptive_versus_maladaptive_self-reflection&amp;oldid=2828912 Adaptive versus maladaptive self-reflection]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Introjection_and_guilt-based_motivation&amp;oldid=2828320 Introjection and guilt-based motivation]

;2025
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&amp;oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&amp;oldid=2726043 Elina.jean.r]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&amp;oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&amp;oldid=2727975 U3254978]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&amp;oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&amp;oldid=2739659 U3237996]

;2024
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&amp;oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&amp;oldid=2644098 U3216883]

;2023
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&amp;oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&amp;oldid=2550752 U3227684]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&amp;oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&amp;oldid=2552580 U3223114]
&lt;!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides.

;2022
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&amp;oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&amp;oldid=2420008 U3203545]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&amp;oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&amp;oldid=2419862 U3210431]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&amp;oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&amp;oldid=2420416 U3216256]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&amp;oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&amp;oldid=2419390 Icantchooseone]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&amp;oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&amp;oldid=2420715 U3217287]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&amp;oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&amp;oldid=2418874 Molzaroid]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&amp;oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&amp;oldid=2420232 Ana028]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&amp;oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&amp;oldid=2418246 U3211603]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&amp;oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&amp;oldid=2417630 U3223109]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&amp;oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&amp;oldid=2420956 U3213441]

;2021
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&amp;oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&amp;action=history U3186377]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&amp;oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&amp;action=history U3202904]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&amp;oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&amp;oldid=2313868 U3194166]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&amp;oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&amp;oldid=2313844 U3187741]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&amp;oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&amp;oldid=2310813 U3187874]
* [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&amp;oldid=2312677 U3202854]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&amp;oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&amp;oldid=2311466 U3187178]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&amp;oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&amp;oldid=2311388 J.Payten]

;2020
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&amp;oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&amp;oldid=2191350 U3170940]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&amp;oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&amp;oldid=2198005 U3190210]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&amp;oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&amp;oldid=2198079 U3160224]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&amp;oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&amp;oldid=2194797 U3190467]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&amp;oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&amp;oldid=2199869 NUMBLA0371]
;2019
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&amp;oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&amp;oldid=2051716 U3172958]
;2018
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&amp;oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&amp;oldid=1915671 U3160212]
;2017
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&amp;oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&amp;oldid=1730836 U3122707]
--&gt;

==Licensing==
Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener.

==See also==
* Structure
** [[Template:Motivation and emotion/Book chapter structure|Book chapter structure template]]
** [[/Checklist|Topic development — Checklist]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]]
** [[Template:METF|Feedback template]]
** [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]
* Tutorials
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 1: Topic selection]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2: Wiki editing]]
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Topic| ]]
[[Category:Motivation and emotion guidelines]]</text>
      <sha1>8jtb6a10pvx1yeghekpr2amfqfuhui5</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Assessment/Topic/Feedback</title>
    <ns>0</ns>
    <id>228705</id>
    <revision>
      <id>2834729</id>
      <parentid>2740565</parentid>
      <timestamp>2026-09-27T22:32:03Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Update for 2026</comment>
      <origin>2834729</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5896" sha1="t9fefkwwb7w0gmux7ubv79unataevxe" xml:space="preserve">{{title|Topic development - General feedback}}
This page summarises general feedback about the [[Motivation and emotion/Assessment/Topic|topic development]] submissions. Read this general feedback in conjunction with individual feedback placed on each topic's discussion page.

==Title and sub-title==
# The title and sub-title should match the ''exact'' wording and casing listed in the [[Motivation and emotion/Book|book's table of contents]]; any changes should be negotiated.
# Don't display user name – authorship is as per the page's editing history (there are likely multiple contributors).

== Headings ==
# The best structures are usually 2-levels with 3 to 6 main body top-level headings between the Overview and Conclusion and, for longer sections, 2 to 5 sub-headings.
# The best structures exhibit close alignment between the sub-title question(s), focus questions, and top-level headings.
# Stronger topic development proposals tend to use more descriptives headings, however aren't overly complicated. Humans tend to create overly short headings whereas genAI [[w:Large language model|LLM]] tools tend to create overly long headings.
# Headings should use sentence casing (i.e., lower-case except for the first letter and proper nouns).

==Overview==
# The best topic developments start with a short scenario and corresponding figure in a feature box, then briefly describe the problem/phenomenon from a psychological point of view, and finish with well-honed focus questions that unpack the sub-title in a feature box.

== Key points ==
# The best submissions develop at least three key points per section with citations to the best psychological theory and research about the topic.
# Submissions with limited or missing development of key points indicate insufficient understanding of the topic.
# A plan for the Conclusion (the most important section) was often missing or underdeveloped.
# When a section has sub-sections, provide an introductory paragraph before the first sub-heading.
# Many submissions could be improved by using [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity], [https://www.grammarly.com/ Grammarly], and/or other tools such as [[Motivation and emotion/Assessment/Using generative AI|generative artificial intelligence]] (genAI) large language models to correct grammatical and spelling errors and to improve the quality of written expression.
# However, overreliance on genAI commonly leads to issues such as overly complicated heading structures, overly broad and overly specific focus questions, overly comprehensive details, and plans that do not show that the author has an indepth understanding of the best psychological theory and research about the topic.
# Use of [[Motivation and emotion/Assessment/Using generative AI|genAI content]] needs to be acknowledged in edit summaries, otherwise it risks triggering learning validation conversations and referrals for summary inquiries for violating principles of academic integrity.

== Figure ==
# Almost all topic developments included a relevant image.
# Most figure captions could be improved by expanding them to a make clearer connection to key points in the main text.
# Cite each figure at least once using APA style (e.g., see Figure 1).

== Learning feature ==
# The best topic developments use at least one out of the following learning features:
## multiple embedded [[m:Help:Interwiki linking|interwiki links]] for the first mention of key words to relevant book chapters and/or Wikipedia articles
## examples/scenarios/case studies (often more examples would improve the chapter). The best scenarios show theory in action through real-world type scenarios.
## figures
## quiz questions about the take-away message(s) (avoid placing these in a stand-alone section at the end – instead, embed each question(s) within the relevant section)

== References ==
# Very few submissions use perfect APA style (7th ed.) for citations and references.
# The most common issues are:
## incorrect capitalisation
## incorrect italicisation
## providing dois as plain text rather than as active hyperlinks
# The best submissions include relevant systematic reviews or meta-analyses.

== Resources ==
# See also
## The best topic developments provide interwiki links to at least one Wikipedia and at least one Wikiversity page, using bullet points, with source  information in brackets - e.g., (Wikipedia) or (Book chapter, 2025).
# External links
## Include source in brackets after the link - e.g., (YouTube).
## Ensure relevance to an ''international'' audience.

== User page ==
# Generally used effectively, with a self-introduction and link to book chapter.
# The main area for potential improvement are expanding information about the user.
# Consider linking to your professional online profile(s) — the most effective professional networking strategy is to do interesting things and share them publicly[https://x.com/JamesClear/status/1958951049806434751].

== Social contribution ==
# Sometimes three of the ''same'' type of contributions are made rather than demonstrating an ability to make three ''different'' types of contribution:
## Direct edit to improve a past or present chapter.
## Comment on the talk page of a past or present chapter.
## Post to an external discussion platform about the topic or unit (e.g., UCLearn discussion of #emot24 on X).
# Only ''direct links to evidence'' are counted for marking purposes. To create a direct link to Wikiversity edits, view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and then copy this website address to your user page. For more info, see the [[Motivation_and_emotion/Assessment/Chapter#Social contribution (10%)|book chapter author guidelines]].

[[Category:Motivation and emotion/Assessment/Topic]]</text>
      <sha1>t9fefkwwb7w0gmux7ubv79unataevxe</sha1>
    </revision>
  </page>
  <page>
    <title>Plant Divisions (Phyla)</title>
    <ns>0</ns>
    <id>235272</id>
    <revision>
      <id>2834732</id>
      <parentid>2833822</parentid>
      <timestamp>2026-09-27T23:04:12Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <origin>2834732</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6333" sha1="8l39uq9fly56xdk64pkmgrnefm1lhsq" xml:space="preserve">[[Image:Diversity of plants (Streptophyta) version 2.png|thumb|300px|right|A sample of plant diversity.]]

In botany, the equivalent of a Phylum is called a division. The Kingdom Plantae is divided into 13 Divisions. A Division (pl. Phyla) is the largest formal major grouping within plant taxonomy below kingdoms. 

This list is presented in alphabetical order, and not in any systematic/evolutionary arrangement.

Science is by no means static. There are arguments of all sizes and shapes about the taxonomy of the Plant Divisions. Other sources may combine or split these listed Divisions. However, at this time, the list presented here should stand in good stead for an introduction to the topic of plant diversity.

There are approximately 380,000 plant species that have been described by science.

This list tries to give the following information on each Division:
*Division Name
*A link to a subpage discussing that Phylum in more detail (if it yet exists)
*Name Meaning (in English)
*An English Common Name, where one is in regular use
*Distinguishing characteristics of plants within the Division
*An approximate number of species described within that Division. Since botany does not stand still, this number can change.

You can also see [[Introduction to Taxonomy]] for more on that topic.

==Anthocerotophyta==
[[Image:Hornwort (3144429129).jpg|thumb|100px|right|Hornworts.]]
[[/Anthocerotophyta/]]

Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300 or more
==Bryophyta==
[[Image:Mose09.jpg|thumb|100px|right|Moss, a bryophyte.]]
[[/Bryophyta/]]

Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

==Charophyta==
[[Image:Chara overview.jpg|thumb|100px|right|''Chara'', a Charophyte.]]
[[/Charophyta/]]

Name Meaning: Chara-like plant

English Common Name: Charophytes

Major distinguishing characteristics:

Approximate number of species described: 1,000

==Chlorophyta==
[[image:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|100px|right|A Chlorophyte.]]
[[/Chlorophyta/]]

Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000

==Cycadophyta==
[[Image:Unidentified cycad in greenhouse.jpg|thumb|100px|right|Unidentified cycad in greenhouse.]]
[[/Cycadophyta/]]

Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

==Ginkgophyta==
[[Image:Gingko biloba2.jpg|thumb|100px|right|''Gingko biloba''.]]
[[/Ginkgophyta/]]

Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

==Glaucophyta==
[[Image:Glaucophyte.jpg|thumb|100px|right|A glaucophyte.]]
[[/Glaucophyta/]] 

Name Meaning: Blue-green plant

English Common Name: Glaucophytes

Major distinguishing characteristics:

Approximate number of species described: 13

==Gnetophyta==
[[Image:Gnetum scandens (6780786863).jpg|thumb|100px|right|''Gnetum scandens''.]]
[[Gnetophyta]]

Name Meaning: Gnetum-like plant

English Common Name: Gnetophytes

Major distinguishing characteristics: Seeds and woody vascular system with vessels.

Approximate number of species described: 70

==Lycopodiophyta (Lycophyta)==
[[Image:Clubmoss - Flickr - pellaea (1).jpg|thumb|100px|right|Clubmoss.]]
[[/Lycopodiophyta (Lycophyta)/]]

Name Meaning: Lycopodium-like plants, wolf plant

English Common Name: Clubmosses, spikemosses

Major distinguishing characteristics: Microphyll leaves, vascular system

Approximate number of species described: 1290 living

==Magnoliophyta (Anthophyta)==
[[Image:Sweetbay Magnolia Magnolia virginiana Flower Closeup 2242px.jpg|thumb|100px|right|''Magnolia virginiana''.]]
[[Plant Divisions (Phyla)/Magnoliophyta|Magnoliophyta]]

Name Meaning: Magnolia-like plant

English Common Name: Flowering plants, angiosperms

Major distinguishing characteristics: Flowers and fruit, vascular system with vessels

Approximate number of species described: 300,000

==Marchantiophyta (Hepatophyta)==
[[Image:Liverwort Ferndale Park.jpg|thumb|100px|right|Liverwort.]]
[[Plant Divisions (Phyla)/Marchantiophyta|Marchantiophyta (Hepatophyta)]]

Name Meaning: Marchantia-like plant, liver plant

English Common Name: Liverworts

Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system

Approximate number of species described: 9,000

==Pinophyta (Coniferophyta)==
[[Image:Taxus wallichiana kz1.jpg|thumb|100px|right|''Taxus wallichiana'', the Himalayan Yew, a conifer.]]
[[/Pinophyta (Coniferophyta)/]]

Name Meaning: Pinus-like plant, cone-bearing plant

English Common Name: Conifers

Major distinguishing characteristics: Cones containing seeds and wood composed of tracheids

Approximate number of species described: 629 extant

==Polypodiophyta (Monilophyta)==
[[Image:Tree Fern.jpg|thumb|100px|right|Tree fern fronds and fiddleneck (growing young frond).]]
[[wikipedia:Fern|Polypodiophyta]] (Monophyte) 

Once called [[wikipedia:Fern|Pteridophyta]] (outdated! The sub-divisions Lycopodiophyte and Euphyllophyte have been differentiated) 

Name Meaning: Many foot plant, Polypodium-like plant

English Common Name: ferns, horsetails

Major distinguishing characteristics: Prothallus gametophytes and vascular system

Approximate number of species described: 9000

==Other Resources==
*[http://tolweb.org/Green_plants/2382 Tree of Life, Green Plants]
*[http://eol.org/pages/281/overview Encyclopedia of life, Plantae]
*[[Animal Phyla]] a companion piece to this one 

==Quizzes==
[[Plant Divisions (Phyla)/Magnoliophyta/Quiz]]
==References==
* [[Wikipedia:Phylum]]
{{reflist}}
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>8l39uq9fly56xdk64pkmgrnefm1lhsq</sha1>
    </revision>
    <revision>
      <id>2834733</id>
      <parentid>2834732</parentid>
      <timestamp>2026-09-27T23:05:49Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>/* Glaucophyta */</comment>
      <origin>2834733</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6484" sha1="7kd7xp8r76smlwjz2mzp6cchm86hw93" xml:space="preserve">[[Image:Diversity of plants (Streptophyta) version 2.png|thumb|300px|right|A sample of plant diversity.]]

In botany, the equivalent of a Phylum is called a division. The Kingdom Plantae is divided into 13 Divisions. A Division (pl. Phyla) is the largest formal major grouping within plant taxonomy below kingdoms. 

This list is presented in alphabetical order, and not in any systematic/evolutionary arrangement.

Science is by no means static. There are arguments of all sizes and shapes about the taxonomy of the Plant Divisions. Other sources may combine or split these listed Divisions. However, at this time, the list presented here should stand in good stead for an introduction to the topic of plant diversity.

There are approximately 380,000 plant species that have been described by science.

This list tries to give the following information on each Division:
*Division Name
*A link to a subpage discussing that Phylum in more detail (if it yet exists)
*Name Meaning (in English)
*An English Common Name, where one is in regular use
*Distinguishing characteristics of plants within the Division
*An approximate number of species described within that Division. Since botany does not stand still, this number can change.

You can also see [[Introduction to Taxonomy]] for more on that topic.

==Anthocerotophyta==
[[Image:Hornwort (3144429129).jpg|thumb|100px|right|Hornworts.]]
[[/Anthocerotophyta/]]

Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300 or more
==Bryophyta==
[[Image:Mose09.jpg|thumb|100px|right|Moss, a bryophyte.]]
[[/Bryophyta/]]

Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

==Charophyta==
[[Image:Chara overview.jpg|thumb|100px|right|''Chara'', a Charophyte.]]
[[/Charophyta/]]

Name Meaning: Chara-like plant

English Common Name: Charophytes

Major distinguishing characteristics:

Approximate number of species described: 1,000

==Chlorophyta==
[[image:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|100px|right|A Chlorophyte.]]
[[/Chlorophyta/]]

Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000

==Cycadophyta==
[[Image:Unidentified cycad in greenhouse.jpg|thumb|100px|right|Unidentified cycad in greenhouse.]]
[[/Cycadophyta/]]

Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

==Ginkgophyta==
[[Image:Gingko biloba2.jpg|thumb|100px|right|''Gingko biloba''.]]
[[/Ginkgophyta/]]

Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

==Glaucophyta==
[[Image:Glaucophyte.jpg|thumb|100px|right|A glaucophyte.]]
[[/Glaucophyta/]] 

Name Meaning: Blue-green plant

English Common Name: Glaucophytes

Major distinguishing characteristics:

Approximate number of species described: 14 - 26&lt;ref&gt;[http://jcs.biologists.org/content/131/2/jcs203414 The monoplastidic bottleneck in algae and plant evolution | Journal of Cell Science]&lt;/ref&gt;

==Gnetophyta==
[[Image:Gnetum scandens (6780786863).jpg|thumb|100px|right|''Gnetum scandens''.]]
[[Gnetophyta]]

Name Meaning: Gnetum-like plant

English Common Name: Gnetophytes

Major distinguishing characteristics: Seeds and woody vascular system with vessels.

Approximate number of species described: 70

==Lycopodiophyta (Lycophyta)==
[[Image:Clubmoss - Flickr - pellaea (1).jpg|thumb|100px|right|Clubmoss.]]
[[/Lycopodiophyta (Lycophyta)/]]

Name Meaning: Lycopodium-like plants, wolf plant

English Common Name: Clubmosses, spikemosses

Major distinguishing characteristics: Microphyll leaves, vascular system

Approximate number of species described: 1290 living

==Magnoliophyta (Anthophyta)==
[[Image:Sweetbay Magnolia Magnolia virginiana Flower Closeup 2242px.jpg|thumb|100px|right|''Magnolia virginiana''.]]
[[Plant Divisions (Phyla)/Magnoliophyta|Magnoliophyta]]

Name Meaning: Magnolia-like plant

English Common Name: Flowering plants, angiosperms

Major distinguishing characteristics: Flowers and fruit, vascular system with vessels

Approximate number of species described: 300,000

==Marchantiophyta (Hepatophyta)==
[[Image:Liverwort Ferndale Park.jpg|thumb|100px|right|Liverwort.]]
[[Plant Divisions (Phyla)/Marchantiophyta|Marchantiophyta (Hepatophyta)]]

Name Meaning: Marchantia-like plant, liver plant

English Common Name: Liverworts

Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system

Approximate number of species described: 9,000

==Pinophyta (Coniferophyta)==
[[Image:Taxus wallichiana kz1.jpg|thumb|100px|right|''Taxus wallichiana'', the Himalayan Yew, a conifer.]]
[[/Pinophyta (Coniferophyta)/]]

Name Meaning: Pinus-like plant, cone-bearing plant

English Common Name: Conifers

Major distinguishing characteristics: Cones containing seeds and wood composed of tracheids

Approximate number of species described: 629 extant

==Polypodiophyta (Monilophyta)==
[[Image:Tree Fern.jpg|thumb|100px|right|Tree fern fronds and fiddleneck (growing young frond).]]
[[wikipedia:Fern|Polypodiophyta]] (Monophyte) 

Once called [[wikipedia:Fern|Pteridophyta]] (outdated! The sub-divisions Lycopodiophyte and Euphyllophyte have been differentiated) 

Name Meaning: Many foot plant, Polypodium-like plant

English Common Name: ferns, horsetails

Major distinguishing characteristics: Prothallus gametophytes and vascular system

Approximate number of species described: 9000

==Other Resources==
*[http://tolweb.org/Green_plants/2382 Tree of Life, Green Plants]
*[http://eol.org/pages/281/overview Encyclopedia of life, Plantae]
*[[Animal Phyla]] a companion piece to this one 

==Quizzes==
[[Plant Divisions (Phyla)/Magnoliophyta/Quiz]]
==References==
* [[Wikipedia:Phylum]]
{{reflist}}
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>7kd7xp8r76smlwjz2mzp6cchm86hw93</sha1>
    </revision>
    <revision>
      <id>2834734</id>
      <parentid>2834733</parentid>
      <timestamp>2026-09-27T23:07:04Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>/* Other Resources */</comment>
      <origin>2834734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6529" sha1="2x1yemeo2r579zsrp0q0fnkwr9k0vm6" xml:space="preserve">[[Image:Diversity of plants (Streptophyta) version 2.png|thumb|300px|right|A sample of plant diversity.]]

In botany, the equivalent of a Phylum is called a division. The Kingdom Plantae is divided into 13 Divisions. A Division (pl. Phyla) is the largest formal major grouping within plant taxonomy below kingdoms. 

This list is presented in alphabetical order, and not in any systematic/evolutionary arrangement.

Science is by no means static. There are arguments of all sizes and shapes about the taxonomy of the Plant Divisions. Other sources may combine or split these listed Divisions. However, at this time, the list presented here should stand in good stead for an introduction to the topic of plant diversity.

There are approximately 380,000 plant species that have been described by science.

This list tries to give the following information on each Division:
*Division Name
*A link to a subpage discussing that Phylum in more detail (if it yet exists)
*Name Meaning (in English)
*An English Common Name, where one is in regular use
*Distinguishing characteristics of plants within the Division
*An approximate number of species described within that Division. Since botany does not stand still, this number can change.

You can also see [[Introduction to Taxonomy]] for more on that topic.

==Anthocerotophyta==
[[Image:Hornwort (3144429129).jpg|thumb|100px|right|Hornworts.]]
[[/Anthocerotophyta/]]

Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300 or more
==Bryophyta==
[[Image:Mose09.jpg|thumb|100px|right|Moss, a bryophyte.]]
[[/Bryophyta/]]

Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

==Charophyta==
[[Image:Chara overview.jpg|thumb|100px|right|''Chara'', a Charophyte.]]
[[/Charophyta/]]

Name Meaning: Chara-like plant

English Common Name: Charophytes

Major distinguishing characteristics:

Approximate number of species described: 1,000

==Chlorophyta==
[[image:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|100px|right|A Chlorophyte.]]
[[/Chlorophyta/]]

Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000

==Cycadophyta==
[[Image:Unidentified cycad in greenhouse.jpg|thumb|100px|right|Unidentified cycad in greenhouse.]]
[[/Cycadophyta/]]

Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

==Ginkgophyta==
[[Image:Gingko biloba2.jpg|thumb|100px|right|''Gingko biloba''.]]
[[/Ginkgophyta/]]

Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

==Glaucophyta==
[[Image:Glaucophyte.jpg|thumb|100px|right|A glaucophyte.]]
[[/Glaucophyta/]] 

Name Meaning: Blue-green plant

English Common Name: Glaucophytes

Major distinguishing characteristics:

Approximate number of species described: 14 - 26&lt;ref&gt;[http://jcs.biologists.org/content/131/2/jcs203414 The monoplastidic bottleneck in algae and plant evolution | Journal of Cell Science]&lt;/ref&gt;

==Gnetophyta==
[[Image:Gnetum scandens (6780786863).jpg|thumb|100px|right|''Gnetum scandens''.]]
[[Gnetophyta]]

Name Meaning: Gnetum-like plant

English Common Name: Gnetophytes

Major distinguishing characteristics: Seeds and woody vascular system with vessels.

Approximate number of species described: 70

==Lycopodiophyta (Lycophyta)==
[[Image:Clubmoss - Flickr - pellaea (1).jpg|thumb|100px|right|Clubmoss.]]
[[/Lycopodiophyta (Lycophyta)/]]

Name Meaning: Lycopodium-like plants, wolf plant

English Common Name: Clubmosses, spikemosses

Major distinguishing characteristics: Microphyll leaves, vascular system

Approximate number of species described: 1290 living

==Magnoliophyta (Anthophyta)==
[[Image:Sweetbay Magnolia Magnolia virginiana Flower Closeup 2242px.jpg|thumb|100px|right|''Magnolia virginiana''.]]
[[Plant Divisions (Phyla)/Magnoliophyta|Magnoliophyta]]

Name Meaning: Magnolia-like plant

English Common Name: Flowering plants, angiosperms

Major distinguishing characteristics: Flowers and fruit, vascular system with vessels

Approximate number of species described: 300,000

==Marchantiophyta (Hepatophyta)==
[[Image:Liverwort Ferndale Park.jpg|thumb|100px|right|Liverwort.]]
[[Plant Divisions (Phyla)/Marchantiophyta|Marchantiophyta (Hepatophyta)]]

Name Meaning: Marchantia-like plant, liver plant

English Common Name: Liverworts

Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system

Approximate number of species described: 9,000

==Pinophyta (Coniferophyta)==
[[Image:Taxus wallichiana kz1.jpg|thumb|100px|right|''Taxus wallichiana'', the Himalayan Yew, a conifer.]]
[[/Pinophyta (Coniferophyta)/]]

Name Meaning: Pinus-like plant, cone-bearing plant

English Common Name: Conifers

Major distinguishing characteristics: Cones containing seeds and wood composed of tracheids

Approximate number of species described: 629 extant

==Polypodiophyta (Monilophyta)==
[[Image:Tree Fern.jpg|thumb|100px|right|Tree fern fronds and fiddleneck (growing young frond).]]
[[wikipedia:Fern|Polypodiophyta]] (Monophyte) 

Once called [[wikipedia:Fern|Pteridophyta]] (outdated! The sub-divisions Lycopodiophyte and Euphyllophyte have been differentiated) 

Name Meaning: Many foot plant, Polypodium-like plant

English Common Name: ferns, horsetails

Major distinguishing characteristics: Prothallus gametophytes and vascular system

Approximate number of species described: 9000

==Other Resources==
*[http://tolweb.org/Green_plants/2382 Tree of Life, Green Plants]
*[http://eol.org/pages/281/overview Encyclopedia of life, Plantae]
*[[Animal Phyla]], a companion piece to this one
*[[Streptophytes]], a &quot;slice&quot; of this page.*

==Quizzes==
[[Plant Divisions (Phyla)/Magnoliophyta/Quiz]]
==References==
* [[Wikipedia:Phylum]]
{{reflist}}
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>2x1yemeo2r579zsrp0q0fnkwr9k0vm6</sha1>
    </revision>
    <revision>
      <id>2834753</id>
      <parentid>2834734</parentid>
      <timestamp>2026-09-28T00:57:13Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <minor />
      <comment>/* Gnetophyta */</comment>
      <origin>2834753</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6531" sha1="it2sxm2p6ruqornrtkayy6rd879nkkr" xml:space="preserve">[[Image:Diversity of plants (Streptophyta) version 2.png|thumb|300px|right|A sample of plant diversity.]]

In botany, the equivalent of a Phylum is called a division. The Kingdom Plantae is divided into 13 Divisions. A Division (pl. Phyla) is the largest formal major grouping within plant taxonomy below kingdoms. 

This list is presented in alphabetical order, and not in any systematic/evolutionary arrangement.

Science is by no means static. There are arguments of all sizes and shapes about the taxonomy of the Plant Divisions. Other sources may combine or split these listed Divisions. However, at this time, the list presented here should stand in good stead for an introduction to the topic of plant diversity.

There are approximately 380,000 plant species that have been described by science.

This list tries to give the following information on each Division:
*Division Name
*A link to a subpage discussing that Phylum in more detail (if it yet exists)
*Name Meaning (in English)
*An English Common Name, where one is in regular use
*Distinguishing characteristics of plants within the Division
*An approximate number of species described within that Division. Since botany does not stand still, this number can change.

You can also see [[Introduction to Taxonomy]] for more on that topic.

==Anthocerotophyta==
[[Image:Hornwort (3144429129).jpg|thumb|100px|right|Hornworts.]]
[[/Anthocerotophyta/]]

Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300 or more
==Bryophyta==
[[Image:Mose09.jpg|thumb|100px|right|Moss, a bryophyte.]]
[[/Bryophyta/]]

Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

==Charophyta==
[[Image:Chara overview.jpg|thumb|100px|right|''Chara'', a Charophyte.]]
[[/Charophyta/]]

Name Meaning: Chara-like plant

English Common Name: Charophytes

Major distinguishing characteristics:

Approximate number of species described: 1,000

==Chlorophyta==
[[image:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|100px|right|A Chlorophyte.]]
[[/Chlorophyta/]]

Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000

==Cycadophyta==
[[Image:Unidentified cycad in greenhouse.jpg|thumb|100px|right|Unidentified cycad in greenhouse.]]
[[/Cycadophyta/]]

Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

==Ginkgophyta==
[[Image:Gingko biloba2.jpg|thumb|100px|right|''Gingko biloba''.]]
[[/Ginkgophyta/]]

Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

==Glaucophyta==
[[Image:Glaucophyte.jpg|thumb|100px|right|A glaucophyte.]]
[[/Glaucophyta/]] 

Name Meaning: Blue-green plant

English Common Name: Glaucophytes

Major distinguishing characteristics:

Approximate number of species described: 14 - 26&lt;ref&gt;[http://jcs.biologists.org/content/131/2/jcs203414 The monoplastidic bottleneck in algae and plant evolution | Journal of Cell Science]&lt;/ref&gt;

==Gnetophyta==
[[Image:Gnetum scandens (6780786863).jpg|thumb|100px|right|''Gnetum scandens''.]]
[[/Gnetophyta/]]

Name Meaning: Gnetum-like plant

English Common Name: Gnetophytes

Major distinguishing characteristics: Seeds and woody vascular system with vessels.

Approximate number of species described: 70

==Lycopodiophyta (Lycophyta)==
[[Image:Clubmoss - Flickr - pellaea (1).jpg|thumb|100px|right|Clubmoss.]]
[[/Lycopodiophyta (Lycophyta)/]]

Name Meaning: Lycopodium-like plants, wolf plant

English Common Name: Clubmosses, spikemosses

Major distinguishing characteristics: Microphyll leaves, vascular system

Approximate number of species described: 1290 living

==Magnoliophyta (Anthophyta)==
[[Image:Sweetbay Magnolia Magnolia virginiana Flower Closeup 2242px.jpg|thumb|100px|right|''Magnolia virginiana''.]]
[[Plant Divisions (Phyla)/Magnoliophyta|Magnoliophyta]]

Name Meaning: Magnolia-like plant

English Common Name: Flowering plants, angiosperms

Major distinguishing characteristics: Flowers and fruit, vascular system with vessels

Approximate number of species described: 300,000

==Marchantiophyta (Hepatophyta)==
[[Image:Liverwort Ferndale Park.jpg|thumb|100px|right|Liverwort.]]
[[Plant Divisions (Phyla)/Marchantiophyta|Marchantiophyta (Hepatophyta)]]

Name Meaning: Marchantia-like plant, liver plant

English Common Name: Liverworts

Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system

Approximate number of species described: 9,000

==Pinophyta (Coniferophyta)==
[[Image:Taxus wallichiana kz1.jpg|thumb|100px|right|''Taxus wallichiana'', the Himalayan Yew, a conifer.]]
[[/Pinophyta (Coniferophyta)/]]

Name Meaning: Pinus-like plant, cone-bearing plant

English Common Name: Conifers

Major distinguishing characteristics: Cones containing seeds and wood composed of tracheids

Approximate number of species described: 629 extant

==Polypodiophyta (Monilophyta)==
[[Image:Tree Fern.jpg|thumb|100px|right|Tree fern fronds and fiddleneck (growing young frond).]]
[[wikipedia:Fern|Polypodiophyta]] (Monophyte) 

Once called [[wikipedia:Fern|Pteridophyta]] (outdated! The sub-divisions Lycopodiophyte and Euphyllophyte have been differentiated) 

Name Meaning: Many foot plant, Polypodium-like plant

English Common Name: ferns, horsetails

Major distinguishing characteristics: Prothallus gametophytes and vascular system

Approximate number of species described: 9000

==Other Resources==
*[http://tolweb.org/Green_plants/2382 Tree of Life, Green Plants]
*[http://eol.org/pages/281/overview Encyclopedia of life, Plantae]
*[[Animal Phyla]], a companion piece to this one
*[[Streptophytes]], a &quot;slice&quot; of this page.*

==Quizzes==
[[Plant Divisions (Phyla)/Magnoliophyta/Quiz]]
==References==
* [[Wikipedia:Phylum]]
{{reflist}}
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>it2sxm2p6ruqornrtkayy6rd879nkkr</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Bnhassin</title>
    <ns>3</ns>
    <id>250786</id>
    <revision>
      <id>2834834</id>
      <parentid>2834671</parentid>
      <timestamp>2026-09-28T10:49:10Z</timestamp>
      <contributor>
        <username>MediaWiki message delivery</username>
        <id>983498</id>
      </contributor>
      <comment>/* Tech News: 2026-40 */ new section</comment>
      <origin>2834834</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1150619" sha1="spf1wqu3bjdc9im27bdk80v7sx2krh3" xml:space="preserve">== First Message Posting ==

Update Talk on Wikiversity [[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 21:04, 29 June 2019 (UTC)

== Update Sandbox User ==

==  Posting to sandbox == 

Update Sandbox on Wikiversity [[User:Bnhassin/sandbox]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]])[[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 11:04, 25 October 2020 (UTC)

== [[m:Special:MyLanguage/Tech/News/2020/50|Tech News: 2020-50]] ==

&lt;section begin=&quot;technews-2020-W50&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/50|Translations]] are available.

'''Recent changes'''
* You can now put pages on your watchlist for a limited period of time. Some wikis already had this function. [https://meta.wikimedia.org/wiki/Community_Tech/Watchlist_Expiry][https://www.mediawiki.org/wiki/Help:Watchlist_expiry]

'''Changes later this week'''
* Information from Wikidata that is used on a wiki page can be shown in recent changes and watchlists on a Wikimedia wiki. To see this you need to turn on showing Wikidata edits in your watchlist in the preferences. Changes to the Wikidata description in the language of a Wikimedia wiki will then be shown in recent changes and watchlists. This will not show edits to languages that are not relevant to your wiki. [https://lists.wikimedia.org/pipermail/wikidata/2020-November/014402.html][https://phabricator.wikimedia.org/T191831]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2020-12-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2020-12-09|en}}. It will be on all wikis from {{#time:j xg|2020-12-10|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2021|Community Wishlist Survey]] between 8 December and 21 December. The survey decides what the [[m:Community Tech|Community Tech team]] will work on.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/50|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2020-W50&quot;/&gt; 16:15, 7 December 2020 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20754641 --&gt;

== [[m:Special:MyLanguage/Tech/News/2020/51|Tech News: 2020-51]] ==

&lt;section begin=&quot;technews-2020-W51&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/51|Translations]] are available.

'''Recent changes'''
* There is a [[mw:Wikipedia for KaiOS|Wikipedia app]] for [[:w:en:KaiOS|KaiOS]] phones. It was released in India in September. It can now be downloaded in other countries too. [https://diff.wikimedia.org/2020/12/10/growing-wikipedias-reach-with-an-app-for-kaios-feature-phones/]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2020-12-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2020-12-16|en}}. It will be on all wikis from {{#time:j xg|2020-12-17|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/51|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2020-W51&quot;/&gt; 21:34, 14 December 2020 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20803489 --&gt;

== [[m:Special:MyLanguage/Tech/News/2020/52|Tech News: 2020-52]] ==

&lt;section begin=&quot;technews-2020-W52&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/52|Translations]] are available.

'''Tech News'''
* Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 11 January 2021.

'''Recent changes'''
* The &lt;code&gt;&lt;nowiki&gt;{{citation needed}}&lt;/nowiki&gt;&lt;/code&gt; template shows when a statement in a Wikipedia article needs a source. If you click on it when you edit with the visual editor there is a popup that explains this. Now it can also show the reason and when it was added. [https://phabricator.wikimedia.org/T270107]

'''Changes later this week'''
* There is no new MediaWiki version this week or next week.

'''Future changes'''
* You can [[m:WMDE Technical Wishes/Geoinformation/Ideas|propose and discuss]] what technical improvements should be done for geographic information. This could be coordinates, maps or other related things.
* Some wikis use [[mw:Writing systems/LanguageConverter|LanguageConverter]] to switch between writing systems or variants of a language. This can only be done for the entire page. There will be a &lt;code&gt;&lt;nowiki&gt;&lt;langconvert&gt;&lt;/nowiki&gt;&lt;/code&gt; tag that can convert a piece of text on a page. [https://phabricator.wikimedia.org/T263082]
*  Oversighters and stewards can hide entries in [[Special:AbuseLog|Special:AbuseLog]]. They can soon hide multiple entries at once using checkboxes. This works like hiding normal edits. It will happen in early January. [https://phabricator.wikimedia.org/T260904]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/52|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2020-W52&quot;/&gt; 20:54, 21 December 2020 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20833836 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/02|Tech News: 2021-02]] ==

&lt;section begin=&quot;technews-2021-W02&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/02|Translations]] are available.

'''Recent changes'''
* You can choose to be reminded when you have not added an edit summary. This can be done in your preferences. This could conflict with the [[:w:en:CAPTCHA|CAPTCHA]]. This has now been fixed. [https://phabricator.wikimedia.org/T12729]
* You can link to specific log entries. You can get these links for example by clicking the timestamps in the log. Until now, such links to private log entries showed no entry even if you had permission to view private log entries. The links now show the entry. [https://phabricator.wikimedia.org/T269761]
* Admins can use the [[:mw:Special:MyLanguage/Extension:AbuseFilter|abuse filter tool]] to automatically prevent bad edits. Three changes happened last week:
** The filter editing interface now shows syntax errors while you type. This is similar to JavaScript pages. It also shows a warning for regular expressions that match the empty string. New warnings will be added later. [https://phabricator.wikimedia.org/T187686]
** [[m:Special:MyLanguage/Meta:Oversighters|Oversighters]] can now hide multiple filter log entries at once using checkboxes on [[Special:AbuseLog]]. This is how the usual revision deletion works. [https://phabricator.wikimedia.org/T260904]
** When a filter matches too many actions after it has been changed it is &quot;throttled&quot;. The most powerful actions are disabled. This is to avoid many editors getting blocked when an administrator made a mistake. The administrator will now get a notification about this &quot;throttle&quot;.
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a new tool to [https://skins.wmflabs.org/?#/add build new skins]. You can also [https://skins.wmflabs.org/?#/ see] existing [[mw:Special:MyLanguage/Manual:Skins|skins]]. You can [[mw:User talk:Jdlrobson|give feedback]]. [https://lists.wikimedia.org/pipermail/wikitech-l/2020-December/094130.html]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Bots using the API no longer watch pages automatically based on account preferences. Setting the &lt;code&gt;watchlist&lt;/code&gt; to &lt;code&gt;watch&lt;/code&gt; will still work. This is to reduce the size of the watchlist data in the database. [https://phabricator.wikimedia.org/T258108]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[mw:Special:MyLanguage/Extension:Scribunto|Scribunto's]] [[:mw:Extension:Scribunto/Lua reference manual#File metadata|file metadata]] now includes length. [https://phabricator.wikimedia.org/T209679]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:w:en:CSS|CSS]] and [[:w:en:JavaScript|JavaScript]] code pages now have link anchors to [https://patchdemo.wmflabs.org/wikis/40e4795d4448b55a6d8c46ff414bcf78/w/index.php/MediaWiki:En.js#L-125 line numbers]. You can use wikilinks like [[:w:en:MediaWiki:Common.js#L-50]]. [https://phabricator.wikimedia.org/T29531]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] There was a [[mw:MediaWiki 1.36/wmf.25|new version]] of MediaWiki last week. You can read [[mw:MediaWiki 1.36/wmf.25/Changelog|a detailed log]] of all 763 changes. Most of them are very small and will not affect you.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-13|en}}. It will be on all wikis from {{#time:j xg|2021-01-14|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/02|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W02&quot;/&gt; 15:42, 11 January 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20950047 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/03|Tech News: 2021-03]] ==

&lt;section begin=&quot;technews-2021-W03&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/03|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-20|en}}. It will be on all wikis from {{#time:j xg|2021-01-21|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* The [[mw:Special:MyLanguage/Growth|Growth team]] plans to add features to [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer tasks/Experiment analysis, November 2020|get more visitors to edit]] to more Wikipedias. You can help [https://translatewiki.net/w/i.php?title=Special:Translate&amp;group=ext-growthexperiments&amp;language=&amp;filter=&amp;action=translate translating the interface].
* You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. [https://phabricator.wikimedia.org/T271791]
* [[m:Special:MyLanguage/MassMessage|MassMessage]] posts could be automatically timestamped in the future. This is because MassMessage senders can now send pages using MassMessage. Pages are more difficult to sign. If there are times when a MassMessage post should not be timestamped you can [[phab:T270435|let the developers know]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/03|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W03&quot;/&gt; 16:10, 18 January 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20974628 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/04|Tech News: 2021-04]] ==

&lt;section begin=&quot;technews-2021-W04&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/04|Translations]] are available.

'''Problems'''
* You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. You will not be able to read or edit [[:wikitech:Main Page|Wikitech]] for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210128T09 {{#time:j xg|2021-01-28|en}} at 09:00 (UTC)]. [https://phabricator.wikimedia.org/T271791][https://phabricator.wikimedia.org/T272388]

'''Changes later this week'''
* [[m:WMDE Technical Wishes/Bracket Matching|Bracket matching]] will be added to the [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] syntax highlighter on the first wikis. The first wikis are German and Catalan Wikipedia and maybe other Wikimedia wikis. This will happen on 27 January. [https://phabricator.wikimedia.org/T270238]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-27|en}}. It will be on all wikis from {{#time:j xg|2021-01-28|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/04|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W04&quot;/&gt; 18:31, 25 January 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21007423 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/05|Tech News: 2021-05]] ==

&lt;section begin=&quot;technews-2021-W05&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/05|Translations]] are available.

'''Problems'''
* [[:w:en:IPv6|IPv6 addresses]] were written in lowercase letters in diffs. This caused dead links since [[Special:Contributions|Special:Contributions]] only accepted uppercase letters for the IPs. This has been fixed. [https://phabricator.wikimedia.org/T272225]

'''Changes later this week'''
* You can soon use Wikidata to link to pages on the multilingual Wikisource. [https://phabricator.wikimedia.org/T138332]
* Often editors use a &quot;non-breaking space&quot; to make a gap between two items when reading but still show them together. This can be used to avoid a line break. You will now be able to add new ones via the special character tool in the 2010, 2017, and visual editors. The character will be shown in the visual editor as a space with a grey background. [https://phabricator.wikimedia.org/T70429][https://phabricator.wikimedia.org/T96666]
* [[File:Octicons-tools.svg|15px|link=| Advanced item]] Wikis use [[mw:Special:MyLanguage/Extension:AbuseFilter|abuse filters]] to stop bad edits being made. Filter maintainers can now use syntax like &lt;code&gt;1.2.3.4 - 1.2.3.55&lt;/code&gt; as well as the &lt;code&gt;1.2.3.4/27&lt;/code&gt; syntax for IP ranges. [https://phabricator.wikimedia.org/T218074]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-03|en}}. It will be on all wikis from {{#time:j xg|2021-02-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* [[mw:Skin:Minerva Neue|Minerva]] is the skin Wikimedia wikis use for mobile traffic. When a page is protected and you can't edit it you can normally read the source wikicode. This doesn't work on Minerva on mobile devices. This is being fixed. Some text might overlap. This is because your community needs to update [[MediaWiki:Protectedpagetext|MediaWiki:Protectedpagetext]] to work on mobile. You can [[phab:T208827|read more]]. [https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Inline_styles_should_not_use_properties_that_impact_sizing_and_positioning][https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Avoid_tables_for_anything_except_data]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] will change the IP address they use to contact the wikis. The new IP address will be &lt;code&gt;185.15.56.1&lt;/code&gt;. This will happen on February 8. You can [[:wikitech:News/CloudVPS NAT wikis|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/05|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W05&quot;/&gt; 22:38, 1 February 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21033195 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/06|Tech News: 2021-06]] ==

&lt;section begin=&quot;technews-2021-W06&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/06|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Wikimedia Apps|Wikipedia app]] for Android now has watchlists and talk pages in the app. [https://play.google.com/store/apps/details?id=org.wikipedia]

'''Changes later this week'''
* You can see edits to chosen pages on [[Special:Watchlist|Special:Watchlist]]. You can add pages to your watchlist on every wiki you like. The [[:mw:Special:MyLanguage/Extension:GlobalWatchlist|GlobalWatchlist]] extension will come to Meta on 11 February. There you can see entries on watched pages on different wikis on the same page. The new watchlist will be found on [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] on Meta. You can choose which wikis to watch and other preferences on [[m:Special:GlobalWatchlistSettings|Special:GlobalWatchlistSettings]] on Meta. You can watch up to five wikis. [https://phabricator.wikimedia.org/T260862]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-10|en}}. It will be on all wikis from {{#time:j xg|2021-02-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* When admins [[mw:Special:MyLanguage/Help:Protecting and unprotecting pages|protect]] pages the form will use the [[mw:UX standardization|OOUI look]]. [[Special:Import|Special:Import]] will also get the new look. This will make them easier to use on mobile phones. [https://phabricator.wikimedia.org/T235424][https://phabricator.wikimedia.org/T108792]
* Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[m:Tech/News/2021/05|Last week]] Tech News reported that the IP address [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] use to contact the wikis will change on 8 February. This is delayed. It will happen later instead. [https://wikitech.wikimedia.org/wiki/News/CloudVPS_NAT_wikis]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/06|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W06&quot;/&gt; 17:42, 8 February 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21082948 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/07|Tech News: 2021-07]] ==

&lt;section begin=&quot;technews-2021-W07&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/07|Translations]] are available.

'''Problems'''
* There were problems with recent versions of MediaWiki.  Because the updates caused problems the developers rolled back to an earlier version. Some updates and new functions will come later than planned. [https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094255.html][https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094271.html]
* Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.31|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-17|en}}. It will be on all wikis from {{#time:j xg|2021-02-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/07|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W07&quot;/&gt; 17:56, 15 February 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21105437 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/08|Tech News: 2021-08]] ==

&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/08|Translations]] are available.

'''Recent changes'''
* The visual editor will now use [[:c:Commons:Structured data/Media search|MediaSearch]] to find images. You can search for images on Commons in the visual editor when you are looking for illustrations. This is to help editors find better images. [https://phabricator.wikimedia.org/T259896]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighter]] now works with more languages: [[:w:en:Futhark (programming language)|Futhark]], [[:w:en:Graphviz|Graphviz]]/[[:w:en:DOT (graph description language)|DOT]], CDDL and AMDGPU. [https://phabricator.wikimedia.org/T274741]

'''Problems'''
* Editing a [[mw:Special:MyLanguage/Extension:EasyTimeline|timeline]] might have removed all text from it. This was because of a bug and has been fixed. You might need to edit the timeline again for it to show properly. [https://phabricator.wikimedia.org/T274822]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.32|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-24|en}}. It will be on all wikis from {{#time:j xg|2021-02-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a [[:m:Wikimedia Rust developers user group|user group]] for developers and users interested in working on Wikimedia wikis with the [[:w:en:Rust (programming language)|Rust programming language]]. You can join or tell others who want to make your wiki better in the future.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;

----

00:17, 23 February 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21134058 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/09|Tech News: 2021-09]] ==

&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/09|Translations]] are available.

'''Recent changes'''
*  Wikis using the [[mw:Special:MyLanguage/Growth/Feature summary|Growth team tools]] can now show the name of a newcomer's mentor anywhere [[mw:Special:MyLanguage/Help:Growth/Mentorship/Integrating_mentorship|through a magic word]]. This can be used for welcome messages or userboxes.
* A new version of the [[c:Special:MyLanguage/Commons:VideoCutTool|VideoCutTool]] is now available. It enables cropping, trimming, audio disabling, and rotating video content. It is being created as part of the developer outreach programs.

'''Problems'''
* There was a problem with the [[mw:Special:MyLanguage/Manual:Job queue|job queue]]. This meant some functions did not save changes and mass messages were delayed. This did not affect wiki edits. [https://phabricator.wikimedia.org/T275437]
* Some editors may not be logged in to their accounts automatically in the latest versions of Firefox and Safari. [https://phabricator.wikimedia.org/T226797]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.33|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-03|en}}. It will be on all wikis from {{#time:j xg|2021-03-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;

----

19:08, 1 March 2021 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21161722 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/10|Tech News: 2021-10]] ==

&lt;section begin=&quot;technews-2021-W10&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/10|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/Content translation/Section translation|Section translation]] now works on Bengali Wikipedia. It helps mobile editors translate sections of articles. It will come to more wikis later. The first focus is active wikis with a smaller number of articles. You can [https://sx.wmflabs.org/index.php/Main_Page test it] and [[mw:Talk:Content translation/Section translation|leave feedback]].
* [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|Flagged revisions]] now give admins the review right. [https://phabricator.wikimedia.org/T275293]
* When someone links to a Wikipedia article on Twitter this will now show a preview of the article. [https://phabricator.wikimedia.org/T276185]

'''Problems'''
* Many graphs have [[:w:en:JavaScript|JavaScript]] errors. Graph editors can check their graphs in their browser's developer console after editing. [https://phabricator.wikimedia.org/T275833]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-10|en}}. It will be on all wikis from {{#time:j xg|2021-03-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).
* The [[mw:Talk pages project/New discussion|New Discussion]] tool will soon be a new [[mw:Special:MyLanguage/Extension:DiscussionTools|discussion tools]] beta feature for on most Wikipedias. The goal is to make it easier to start new discussions. [https://phabricator.wikimedia.org/T275257]

'''Future changes'''
* There will be a number of changes to make it easier to work with templates. Some will come to the first wikis in March. Other changes will come to the first wikis in June. This is both for those who use templates and those who create or maintain them. You can [[:m:WMDE Technical Wishes/Templates|read more]].
* [[m:WMDE Technical Wishes/ReferencePreviews|Reference Previews]] will become a default feature on some wikis on 17 March. They will share a setting with [[mw:Page Previews|Page Previews]]. If you prefer the Reference Tooltips or Navigation-Popups gadget you can keep using them. If so Reference Previews won't be shown. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews]
* New JavaScript-based functions will not work in [[:w:en:Internet Explorer 11|Internet Explorer 11]]. This is because Internet Explorer is an old browser that doesn't work with how JavaScript is written today. Everything that works in Internet Explorer 11 today will continue working in Internet Explorer for now. You can [[mw:Compatibility/IE11|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/10|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W10&quot;/&gt; 17:51, 8 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21175593 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/11|Tech News: 2021-11]] ==

&lt;section begin=&quot;technews-2021-W11&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/11|Translations]] are available.

'''Recent changes'''
* Wikis that are part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|desktop improvements]] project can now use a new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Search|search function]]. The desktop improvements and the new search will come to more wikis later. You can also [[mw:Reading/Web/Desktop Improvements#Deployment plan and timeline|test it early]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Editors who put up banners or change site-wide [[:w:en:JavaScript|JavaScript]] code should use the [https://grafana.wikimedia.org/d/000000566/overview?viewPanel=16&amp;orgId=1 client error graph] to see that their changes has not caused problems. You can [https://diff.wikimedia.org/2021/03/08/sailing-steady%e2%80%8a-%e2%80%8ahow-you-can-help-keep-wikimedia-sites-error-free read more]. [https://phabricator.wikimedia.org/T276296]

'''Problems'''
* Due to [[phab:T276968|database issues]] the [https://meta.wikimedia.beta.wmflabs.org Wikimedia Beta Cluster] was read-only for over a day.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-17|en}}. It will be on all wikis from {{#time:j xg|2021-03-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* You can add a [[:w:en:Newline|newline]] or [[:w:en:Carriage return|carriage return]] character to a custom signature if you use a template. There is a proposal to not allow them in the future. This is because they can cause formatting problems. [https://www.mediawiki.org/wiki/New_requirements_for_user_signatures#Additional_proposal_(2021)][https://phabricator.wikimedia.org/T272322]
* You will be able to read but not edit [[phab:T276899|12 wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|en}} at 06:00 (UTC)]. This could take 30 minutes but will probably be much faster.
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] You can use [https://quarry.wmflabs.org/ Quarry] for [[:w:en:SQL|SQL]] queries to the [[wikitech:Wiki replicas|Wiki Replicas]]. Cross-database &lt;code&gt;JOINS&lt;/code&gt; will no longer work from 23 March. There will be a new field to specify the database to connect to. If you think this affects you and you need help you can [[phab:T268498|post on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]]. [https://wikitech.wikimedia.org/wiki/PAWS PAWS] and other ways to do [[:w:en:SQL|SQL]] queries to the Wiki Replicas will be affected later. [https://wikitech.wikimedia.org/wiki/News/Wiki_Replicas_2020_Redesign]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/11|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W11&quot;/&gt; 23:22, 15 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21226057 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/12|Tech News: 2021-12]] ==

&lt;section begin=&quot;technews-2021-W12&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/12|Translations]] are available.

'''Recent changes'''
* There is a [[mw:Wikipedia for KaiOS|Wikipedia app]] for [[:w:en:KaiOS|KaiOS]] phones. They don't have a touch screen so readers navigate with the phone keys. There is now a [https://wikimedia.github.io/wikipedia-kaios/sim.html simulator] so you can see what it looks like.
* The [[mw:Special:MyLanguage/Talk pages project/Replying|reply tool]] and [[mw:Special:MyLanguage/Talk pages project/New discussion|new discussion tool]] are now available as the &quot;{{int:discussiontools-preference-label}}&quot; [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] in almost all wikis except German Wikipedia.

'''Problems'''
* You will be able to read but not edit [[phab:T276899|twelve wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|{{PAGELANGUAGE}}}} at 06:00 (UTC)]. This can also affect password changes, logging in to new wikis, global renames and changing or confirming emails. This could take 30 minutes but will probably be much faster.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.36|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-24|en}}. It will be on all wikis from {{#time:j xg|2021-03-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).
* [[:w:en:Syntax highlighting|Syntax highlighting]] colours will change to be easier to read. This will soon come to the [[phab:T276346|first wikis]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting]

'''Future changes'''
* [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged revisions]] will no longer have multiple tags like &quot;tone&quot; or &quot;depth&quot;. It will also only have one tier. This was changed because very few wikis used these features and they make the tool difficult to maintain. [https://phabricator.wikimedia.org/T185664][https://phabricator.wikimedia.org/T277883]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets and user scripts can access variables about the current page in JavaScript. In 2015 this was moved from &lt;code dir=ltr&gt;wg*&lt;/code&gt; to &lt;code dir=ltr&gt;mw.config&lt;/code&gt;. &lt;code dir=ltr&gt;wg*&lt;/code&gt; will soon no longer work. [https://phabricator.wikimedia.org/T72470]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/12|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W12&quot;/&gt; 16:53, 22 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21244806 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/13|Tech News: 2021-13]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/13|Translations]] are available.

'''Recent changes'''
* Some very old [[:w:en:Web browser|web browsers]] [[:mw:Special:MyLanguage/Compatibility|don’t work]] well with the Wikimedia wikis. Some old code for browsers that used to be supported is being removed. This could cause issues in those browsers. [https://phabricator.wikimedia.org/T277803]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:m:IRC/Channels#Raw_feeds|IRC recent changes feeds]] have been moved to a new server. Make sure all tools automatically reconnect to &lt;code&gt;irc.wikimedia.org&lt;/code&gt; and not to the name of any specific server. Users should also consider switching to the more modern [[:wikitech:Event Platform/EventStreams|EventStreams]]. [https://phabricator.wikimedia.org/T224579]

'''Problems'''
* When you move a page that many editors have on their watchlist the history can be split. It might also not be possible to move it again for a while. This is because of a [[:w:en:Job queue|job queue]] problem. [https://phabricator.wikimedia.org/T278350]
* Some translatable pages on Meta could not be edited. This was because of a bug in the translation tool. The new MediaWiki version was delayed because of problems like this. [https://phabricator.wikimedia.org/T278429][https://phabricator.wikimedia.org/T274940]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.37|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-31|en}}. It will be on all wikis from {{#time:j xg|2021-04-01|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:30, 29 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21267131 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/14|Tech News: 2021-14]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/14|Translations]] are available.

'''Recent changes'''
* Editors can collapse part of an article so you have to click on it to see it. When you click a link to a section inside collapsed content it will now expand to show the section. The browser will scroll down to the section. Previously such links didn't work unless you manually expanded the content first. [https://phabricator.wikimedia.org/T276741]

'''Changes later this week'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] will use for example &lt;code&gt;2010-12-XX&lt;/code&gt; instead of &lt;code&gt;2010-12&lt;/code&gt; for dates with a month but no days. This is because &lt;code&gt;2010-12&lt;/code&gt; could be confused with &lt;code&gt;2010-2012&lt;/code&gt; instead of &lt;code&gt;December 2010&lt;/code&gt;. This is called level 1 instead of level 0 in the [https://www.loc.gov/standards/datetime/ Extended Date/Time Format]. [https://phabricator.wikimedia.org/T132308]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.38|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-07|en}}. It will be on all wikis from {{#time:j xg|2021-04-08|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:PAWS|PAWS]] can now connect to the new [[:wikitech:Wiki Replicas|Wiki Replicas]]. Cross-database &lt;code&gt;JOINS&lt;/code&gt; will no longer work from 28 April. There is [[:wikitech:News/Wiki Replicas 2020 Redesign#How should I connect to databases in PAWS?|a new way to connect]] to the databases. Until 28 April both ways to connect to the databases will work. If you think this affects you and you need help you can post [[phab:T268498|on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

19:41, 5 April 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21287348 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/16|Tech News: 2021-16]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/16|Translations]] are available.

'''Recent changes'''
* Email to the Wikimedia wikis are handled by groups of Wikimedia editors. These volunteer response teams now use [https://github.com/znuny/Znuny Znuny] instead of [[m:Special:MyLanguage/OTRS|OTRS]]. The functions and interface remain the same. The volunteer administrators will give more details about the next steps soon. [https://phabricator.wikimedia.org/T279303][https://phabricator.wikimedia.org/T275294]
* If you use [[Mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighting]], you can see line numbers in the 2010 and 2017 wikitext editors when editing templates. This is to make it easier to see line breaks or talk about specific lines. Line numbers will soon come to all namespaces. [https://phabricator.wikimedia.org/T267911][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Line_Numbering][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Line_Numbering]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Because of a technical change there could be problems with gadgets and scripts that have an edit summary area that looks [https://phab.wmfusercontent.org/file/data/llvdqqnb5zpsfzylbqcg/PHID-FILE-25vs4qowibmtysl7cbml/Screen_Shot_2021-04-06_at_2.34.04_PM.png similar to this one]. If they look strange they should use &lt;code&gt;mw.loader.using('mediawiki.action.edit.styles')&lt;/code&gt; to go back to how they looked before. [https://phabricator.wikimedia.org/T278898]
* The [[mw:MediaWiki 1.37/wmf.1|latest version]] of MediaWiki came to the Wikimedia wikis last week. There was no Tech News issue last week.

'''Changes later this week'''
*  There is no new MediaWiki version this week.

'''Future changes'''
* The user group &lt;code&gt;oversight&lt;/code&gt; will be renamed &lt;code&gt;suppress&lt;/code&gt;. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in two weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:48, 19 April 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21356080 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/17|Tech News: 2021-17]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/17|Translations]] are available.

'''Recent changes'''
* Templates have parameters that can have specific values. It is possible to suggest values for editors with [[mw:Special:MyLanguage/Extension:TemplateData|TemplateData]]. You can soon see them as a drop-down list in the visual editor. This is to help template users find the right values faster. [https://phabricator.wikimedia.org/T273857][https://meta.wikimedia.org/wiki/Special:MyLanguage/WMDE_Technical_Wishes/Suggested_values_for_template_parameters][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Suggested_values_for_template_parameters]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-28|en}}. It will be on all wikis from {{#time:j xg|2021-04-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

21:24, 26 April 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21391118 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/18|Tech News: 2021-18]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/18|Translations]] are available.

'''Recent changes'''
* [[w:en:Wikipedia:Twinkle|Twinkle]] is a gadget on English Wikipedia. It can help with maintenance and patrolling. It can [[m:Grants:Project/Rapid/SD0001/Twinkle localisation/Report|now be used on other wikis]]. You can get Twinkle on your wiki using the [https://github.com/wikimedia-gadgets/twinkle-starter twinkle-starter] GitHub repository.

'''Problems'''
* The [[mw:Special:MyLanguage/Content translation|content translation tool]] did not work for many articles for a little while. This was because of a bug. [https://phabricator.wikimedia.org/T281346]
* Some things will not work for about a minute on 5 May. This will happen [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210505T0600 around 06:00 UTC]. This will affect the content translation tool and notifications among other things. This is because of an upgrade to avoid crashes. [https://phabricator.wikimedia.org/T281212]

'''Changes later this week'''
* [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] will become a default feature on a number of wikis on 5 May. This is later than planned because of some changes. You can use it without using [[mw:Special:MyLanguage/Page Previews|Page Previews]] if you want to. The earlier plan was to have the preference to use both or none. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-05|en}}. It will be on all wikis from {{#time:j xg|2021-05-06|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:CSS|CSS]] classes &lt;code dir=ltr&gt;.error&lt;/code&gt;, &lt;code dir=ltr&gt;.warning&lt;/code&gt; and &lt;code dir=ltr&gt;.success&lt;/code&gt; do not work for mobile readers if they have not been specifically defined on your wiki. From June they will not work for desktop readers. This can affect gadgets and templates. The classes can be defined in [[MediaWiki:Common.css]] or template styles instead. [https://phabricator.wikimedia.org/T280766]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:43, 3 May 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21418010 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/19|Tech News: 2021-19]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/19|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-12|en}}. It will be on all wikis from {{#time:j xg|2021-05-13|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* You can see what participants plan to work on at the online [[mw:Wikimedia Hackathon 2021|Wikimedia hackathon]] 22–23 May.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:10, 10 May 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21428676 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/20|Tech News: 2021-20]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/20|Translations]] are available.

'''Recent changes'''
* There is a new toolbar in [[mw:Talk pages project/Replying|the Reply tool]]. It works in the wikitext source mode. You can enable it in [[Special:Preferences#mw-htmlform-discussion|your preferences]]. [https://phabricator.wikimedia.org/T276608] [https://www.mediawiki.org/wiki/Talk_pages_project/Replying#13_May_2021] [https://www.mediawiki.org/wiki/Talk_pages_project/New_discussion#13_May_2021]
* Wikimedia [https://lists.wikimedia.org/mailman/listinfo mailing lists] are being moved to [[:w:en:GNU Mailman|Mailman 3]]. This is a newer version. For the [[:w:en:Character encoding|character encoding]] to work it will change from &lt;code&gt;[[:w:en:UTF-8|UTF-8]]&lt;/code&gt; to &lt;code&gt;utf8mb3&lt;/code&gt;. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IEYQ2HS3LZF2P3DAYMNZYQDGHWPVMTPY/][https://phabricator.wikimedia.org/T282621]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] An [[m:Special:MyLanguage/Tech/News/2021/14|earlier issue]] of Tech News said that the  [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] would handle dates with a month but no days in a new way. This has been reverted for now. There needs to be more discussion of how it affects different wikis first. [https://phabricator.wikimedia.org/T132308]

'''Changes later this week'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] &lt;code&gt;MediaWiki:Pageimages-blacklist&lt;/code&gt; will be renamed &lt;code&gt;MediaWiki:Pageimages-denylist&lt;/code&gt;. The list can be copied to the new name. It will happen on 19 May for some wikis and 20 May for some wikis. Most wikis don't use it. It lists images that should never be used as thumbnails for articles. [https://phabricator.wikimedia.org/T282626]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-19|en}}. It will be on all wikis from {{#time:j xg|2021-05-20|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

13:49, 17 May 2021 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21464279 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/21|Tech News: 2021-21]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/21|Translations]] are available.

'''Recent changes'''
* The Wikimedia movement has been using [[:m:Special:MyLanguage/IRC|IRC]] on a network called [[:w:en:Freenode|Freenode]]. There have been changes around who is in control of the network. The [[m:Special:MyLanguage/IRC/Group_Contacts|Wikimedia IRC Group Contacts]] have [[m:Special:Diff/21476411|decided]] to move to the new [[:w:en:Libera Chat|Libera Chat]] network instead. This is not a formal decision for the movement to move all channels but most Wikimedia IRC channels will probably leave Freenode. There is a [[:m:IRC/Migrating_to_Libera_Chat|migration guide]] and ongoing Wikimedia [[m:Wikimedia Forum#Freenode (IRC)|discussions about this]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-26|en}}. It will be on all wikis from {{#time:j xg|2021-05-27|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:07, 24 May 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21477606 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/22|Tech News: 2021-22]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/22|Translations]] are available.

'''Problems'''
* There was an issue on the Vector skin with the text size of categories and notices under the page title. It was fixed last Monday. [https://phabricator.wikimedia.org/T283206]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:05, 31 May 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21516076 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/23|Tech News: 2021-23]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/23|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-09|en}}. It will be on all wikis from {{#time:j xg|2021-06-10|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* The Wikimedia movement uses [[:mw:Special:MyLanguage/Phabricator|Phabricator]] for technical tasks. This is where we collect technical suggestions, bugs and what developers are working on. The company behind Phabricator will stop working on it. This will not change anything for the Wikimedia movement now. It could lead to changes in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/YAXOD46INJLAODYYIJUVQWOZFIV54VUI/][https://admin.phacility.com/phame/post/view/11/phacility_is_winding_down_operations/][https://phabricator.wikimedia.org/T283980]
* Searching on Wikipedia will find more results in some languages. This is mainly true for when those who search do not use the correct [[:w:en:Diacritic|diacritics]] because they are not seen as necessary in that language. For example searching for &lt;code&gt;Bedusz&lt;/code&gt; doesn't find &lt;code&gt;Będusz&lt;/code&gt; on German Wikipedia. The character &lt;code&gt;ę&lt;/code&gt; isn't used in German so many would write &lt;code&gt;e&lt;/code&gt; instead. This will work better in the future in some languages. [https://phabricator.wikimedia.org/T219550]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:Cross-site request forgery|CSRF token parameters]] in the [[:mw:Special:MyLanguage/API:Main page|action API]] were changed in 2014. The old parameters from before 2014 will stop working soon. This can affect bots, gadgets and user scripts that still use the old parameters. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IMP43BNCI32C524O5YCUWMQYP4WVBQ2B/][https://phabricator.wikimedia.org/T280806]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

20:02, 7 June 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21551759 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/24|Tech News: 2021-24]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/24|Translations]] are available.

'''Recent changes'''
* Logged-in users on the mobile web can choose to use the [[:mw:Special:MyLanguage/Reading/Web/Advanced mobile contributions|advanced mobile mode]]. They now see categories in a similar way as users on desktop do. This means that some gadgets that have just been for desktop users could work for users of the mobile site too. If your wiki has such gadgets you could decide to turn them on for the mobile site too. Some gadgets probably need to be fixed to look good on mobile. [https://phabricator.wikimedia.org/T284763]
* Language links on Wikidata now works for [[:oldwikisource:Main Page|multilingual Wikisource]]. [https://phabricator.wikimedia.org/T275958]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''Future changes'''
* In the future we [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation|can't show the IP]] of unregistered editors to everyone. This is because privacy regulations and norms have changed. There is now a rough draft of how [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#Updates|showing the IP to those who need to see it]] could work.
* German Wikipedia, English Wikivoyage and 29 smaller wikis will be read-only for a few minutes on 22 June. This is planned between 5:00 and 5:30 UTC. [https://phabricator.wikimedia.org/T284530]
* All wikis will be read-only for a few minutes in the week of 28 June. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

20:26, 14 June 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21587625 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/25|Tech News: 2021-25]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/25|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The &lt;code&gt;otrs-member&lt;/code&gt; group name is now &lt;code&gt;vrt-permissions&lt;/code&gt;. This could affect abuse filters. [https://phabricator.wikimedia.org/T280615]

'''Problems'''
* You will be able to read but not edit German Wikipedia, English Wikivoyage and 29 smaller wikis for a few minutes on 22 June. This is planned between [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210623T0500 5:00 and 5:30 UTC]. [https://phabricator.wikimedia.org/T284530]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-23|en}}. It will be on all wikis from {{#time:j xg|2021-06-24|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:49, 21 June 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21593987 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/26|Tech News: 2021-26]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/26|Translations]] are available.

'''Recent changes'''
* Wikis with the [[mw:Special:MyLanguage/Growth|Growth features]] now can [[mw:Special:MyLanguage/Growth/Community configuration|configure Growth features directly on their wiki]]. This uses the new special page &lt;code&gt;Special:EditGrowthConfig&lt;/code&gt;. [https://phabricator.wikimedia.org/T285423]
* Wikisources have a new [[m:Special:MyLanguage/Community Tech/OCR Improvements|OCR tool]]. If you don't want to see the &quot;extract text&quot; button on Wikisource you can add &lt;code&gt;.ext-wikisource-ExtractTextWidget { display: none; }&lt;/code&gt; to your [[Special:MyPage/common.css|common.css page]]. [https://phabricator.wikimedia.org/T285311]

'''Problems'''
*You will be able to read but not edit the Wikimedia wikis for a few minutes on 29 June. This is planned at [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210629T1400 14:00 UTC]. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-30|en}}. It will be on all wikis from {{#time:j xg|2021-07-01|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* &lt;code&gt;Threshold for stub link formatting&lt;/code&gt;, &lt;code&gt;thumbnail size&lt;/code&gt; and &lt;code&gt;auto-number headings&lt;/code&gt; can be set in preferences. They are expensive to maintain and few editors use them. The developers are planning to remove them. Removing them will make pages load faster. You can [[mw:Special:MyLanguage/User:SKim (WMF)/Performance Dependent User Preferences|read more and give feedback]].
* A toolbar will be added to the [[mw:Talk pages project/Replying|Reply tool]]'s wikitext source mode. This will make it easier to link to pages and to ping other users. [https://phabricator.wikimedia.org/T276609][https://www.mediawiki.org/wiki/Talk_pages_project/Replying#Status_updates]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:32, 28 June 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21653312 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/27|Tech News: 2021-27]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/27|Translations]] are available.

'''Tech News'''
* The next issue of Tech News will be sent out on 19 July.

'''Recent changes'''
* [[:wikidata:Q4063270|AutoWikiBrowser]] is a tool to make repetitive tasks easier. It now uses [[:w:en:JSON|JSON]]. &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPage&lt;/code&gt; has moved to &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPageJSON&lt;/code&gt; and &lt;code&gt;Wikipedia:AutoWikiBrowser/Config&lt;/code&gt;. &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPage/Version&lt;/code&gt; has moved to &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPage/VersionJSON&lt;/code&gt;. The tool will eventually be configured on the wiki so that you don't have to wait until the new version to add templates or regular expression fixes. [https://phabricator.wikimedia.org/T241196]

'''Problems'''
* [[m:Special:MyLanguage/InternetArchiveBot|InternetArchiveBot]] helps saving online sources on some wikis. It adds them to [[:w:en:Wayback Machine|Wayback Machine]] and links to them there. This is so they don't disappear if the page that was linked to is removed. It currently has a problem with linking to the wrong date when it moves pages from &lt;code&gt;archive.is&lt;/code&gt; to &lt;code&gt;web.archive.org&lt;/code&gt;. [https://phabricator.wikimedia.org/T283432]

'''Changes later this week'''
* The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] will be updated. This is to make it easier to find and use the right templates. It will come to the first wikis on 7 July. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553]
* There is no new MediaWiki version this week.

'''Future changes'''
* Some Wikimedia wikis use [[m:Special:MyLanguage/Flagged Revisions|Flagged Revisions]] or pending changes. It hides edits from new and unregistered accounts for readers until they have been patrolled. The auto review action in Flagged Revisions will no longer be logged. All old logs of auto-review will be removed. This is because it creates a lot of logs that are not very useful. [https://phabricator.wikimedia.org/T285608]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:33, 5 July 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21694636 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/29|Tech News: 2021-29]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/29|Translations]] are available.

'''Recent changes'''
* The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] was updated. This is to make it easier to find and use the right templates. It was supposed to come to the first wikis on 7 July. It was delayed to 12 July instead. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553]
* [[Special:UnconnectedPages|Special:UnconnectedPages]] lists pages that are not connected to Wikidata. This helps you find pages that can be connected to Wikidata items. Some pages should not be connected to Wikidata. You can use the magic word &lt;code&gt;&lt;nowiki&gt;__EXPECTED_UNCONNECTED_PAGE__&lt;/nowiki&gt;&lt;/code&gt; on pages that should not be listed on the special page. [https://phabricator.wikimedia.org/T97577]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-21|en}}. It will be on all wikis from {{#time:j xg|2021-07-22|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] How media is structured in the [[:w:en:Parsing|parser's]] HTML output will soon change. This can affect bots, gadgets, user scripts and extensions. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/L2UQJRHTFK5YG3IOZEC7JSLH2ZQNZRVU/ read more]. You can test it on [[:testwiki:Main Page|Testwiki]] or [[:test2wiki:Main Page|Testwiki 2]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:31, 19 July 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21755027 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/30|Tech News: 2021-30]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/30|Translations]] are available.

'''Recent changes'''
* A [[mw:MediaWiki 1.37/wmf.14|new version]] of MediaWiki came to the Wikimedia wikis the week before last week. This was not in Tech News because there was no newsletter that week. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-28|en}}. It will be on all wikis from {{#time:j xg|2021-07-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* If you use the [[mw:Special:MyLanguage/Skin:MonoBook|Monobook skin]] you can choose to switch off [[:w:en:Responsive web design|responsive design]] on mobile. This will now work for more skins. If &lt;code&gt;{{int:monobook-responsive-label}}&lt;/code&gt; is unticked you need to also untick the new [[Special:Preferences#mw-prefsection-rendering|preference]] &lt;code&gt;{{int:prefs-skin-responsive}}&lt;/code&gt;. Otherwise it will stop working. Interface admins can automate this process on your wiki. You can [[phab:T285991|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

21:11, 26 July 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21771634 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/31|Tech News: 2021-31]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/31|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] If your wiki uses markup like &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-ltr&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-rtl&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; without the required &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;dir&lt;/code&gt;&lt;/bdi&gt; attribute, then these will no longer work in 2 weeks. There is a short-term fix that can be added to your local wiki's Common.css page, which is explained at [[phab:T287701|T287701]]. From now on, all usages should include the full attributes, for example: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; lang=&quot;en&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-rtl&quot; dir=&quot;rtl&quot; lang=&quot;he&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This also applies to some other HTML tags, such as &lt;code&gt;span&lt;/code&gt; or &lt;code&gt;code&lt;/code&gt;. You can find existing examples on your wiki that need to be updated, using the instructions at [[phab:T287701|T287701]].
* Reminder: Wikimedia has [[m:Special:MyLanguage/IRC/Migrating to Libera Chat|migrated to the Libera Chat IRC network]], from the old Freenode network. Local documentation should be updated.

'''Problems'''
* Last week, all wikis had slow access or no access for 30 minutes. There was a problem with generating dynamic lists of articles on the Russian Wikinews, due to the bulk import of 200,000+ new articles over 3 days, which led to database problems. The problematic feature has been disabled on that wiki and developers are discussing if it can be fixed properly. [https://phabricator.wikimedia.org/T287380][https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-07-26_ruwikinews_DynamicPageList]

'''Changes later this week'''
* When adding links to a page using [[mw:VisualEditor|VisualEditor]] or the [[mw:Special:MyLanguage/2017 wikitext editor|2017 wikitext editor]], [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]] will now only appear at the bottom of search results. This is because users do not often want to link to disambiguation pages. [https://phabricator.wikimedia.org/T285510]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-04|en}}. It will be on all wikis from {{#time:j xg|2021-08-05|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* The [[mw:Wikimedia Apps/Team/Android|team of the Wikipedia app for Android]] is working on communication in the app. The developers are working on how to talk to other editors and get notifications. You can [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|read more]]. They are looking for users who want to [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication/UsertestingJuly2021|test the plans]]. Any editor who has an Android phone and is willing to download the app can do this.
* The [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] for {{int:discussiontools-preference-label}} will be updated in the coming weeks.  You will be able to [[mw:Talk pages project/Notifications|subscribe to individual sections]] on a talk page at more wikis.  You can test this now by adding &lt;code&gt;?dtenable=1&lt;/code&gt; to the end of the talk page's URL ([https://meta.wikimedia.org/wiki/Meta_talk:Sandbox?dtenable=1 example]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

20:47, 2 August 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21818289 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/32|Tech News: 2021-32]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/32|Translations]] are available.

'''Problems'''
* You can read but not edit 17 wikis for a few minutes on 10 August. This is planned at [https://zonestamp.toolforge.org/1628571650 05:00 UTC]. This is because of work on the database. [https://phabricator.wikimedia.org/T287449]

'''Changes later this week'''
* The [[wmania:Special:MyLanguage/2021:Hackathon|Wikimania Hackathon]] will take place remotely on 13 August, starting at 5:00 UTC, for 24 hours. You can participate in many ways. You can still propose projects and sessions.
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-11|en}}. It will be on all wikis from {{#time:j xg|2021-08-12|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The old CSS &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;visualClear&quot;&gt;&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will not be supported after 12 August. Instead, templates and pages should use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;clear:both;&quot;&gt;&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Please help to replace any existing uses on your wiki. There are global-search links available at [[phab:T287962|T287962]].

'''Future changes'''
* [[m:Special:MyLanguage/The Wikipedia Library|The Wikipedia Library]] is a place for Wikipedia editors to get access to sources. There is an [[mw:Special:MyLanguage/Extension:TheWikipediaLibrary|extension]] which has a new function to tell users when they can take part in it. It will use notifications. It will start pinging the first users in September. It will ping more users later. [https://phabricator.wikimedia.org/T288070]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[w:en:Vue.js|Vue.js]] will be the [[w:en:JavaScript|JavaScript]] framework for MediaWiki in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/SOZREBYR36PUNFZXMIUBVAIOQI4N7PDU/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:21, 9 August 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21856726 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/33|Tech News: 2021-33]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/33|Translations]] are available.

'''Recent changes'''
* You can add language links in the sidebar in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206]

'''Problems'''
* There was a problem on wikis which use the Translate extension. Translations were not updated or were replaced with the English text. The problems have been fixed. [https://phabricator.wikimedia.org/T288700][https://phabricator.wikimedia.org/T288683][https://phabricator.wikimedia.org/T288719]

'''Changes later this week'''
* A [[mw:Help:Tags|revision tag]] will soon be added to edits that add links to [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]]. This is because these links are usually added by accident. The tag will allow editors to easily find the broken links and fix them. If your wiki does not like this feature, it can be [[mw:Help:Tags#Deleting a tag added by the software|hidden]]. [https://phabricator.wikimedia.org/T287549]
*Would you like to help improve the information about tools? Would you like to attend or help organize a small virtual meetup for your community to discuss the list of tools? Please get in touch on the [[m:Toolhub/The Quality Signal Sessions|Toolhub Quality Signal Sessions]] talk page. We are also looking for feedback [[m:Talk:Toolhub/The Quality Signal Sessions#Discussion topic for &quot;Quality Signal Sessions: The Tool Maintainers edition&quot;|from tool maintainers]] on some specific questions.
* In the past, edits to any page in your user talk space ignored your [[mw:Special:MyLanguage/Help:Notifications#mute|mute list]], e.g. sub-pages. Starting this week, this is only true for edits to your talk page. [https://phabricator.wikimedia.org/T288112]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-18|en}}. It will be on all wikis from {{#time:j xg|2021-08-19|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

19:27, 16 August 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21889213 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/34|Tech News: 2021-34]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/34|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Extension:Score|Score]] extension (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;score&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; notation) has been re-enabled on public wikis and upgraded to a newer version. Some musical score functionality may no longer work because the extension is only enabled in &quot;safe mode&quot;. The security issue has been fixed and an [[mw:Special:MyLanguage/Extension:Score/2021 security advisory|advisory published]].

'''Problems'''
* You will be able to read but not edit [[phab:T289130|some wikis]] for a few minutes on {{#time:j xg|2021-08-25|en}}. This will happen around [https://zonestamp.toolforge.org/1629871217 06:00 UTC]. This is for database maintenance. During this time, operations on the CentralAuth will also not be possible.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-25|en}}. It will be on all wikis from {{#time:j xg|2021-08-26|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

21:58, 23 August 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21923254 --&gt;

== Read-only reminder ==

&lt;section begin=&quot;MassMessage&quot;/&gt;
A maintenance operation will be performed on [https://zonestamp.toolforge.org/1629871231 {{#time: l F d H:i e|2021-08-25T06:00|en}}]. It should only last for a few minutes.

This will affect your wiki as well as 11 other wikis. During this time, publishing edits will not be possible. 

Also during this time, operations on the CentralAuth will not be possible (GlobalRenames, changing/confirming e-mail addresses, logging into new wikis, password changes).

For more details about the operation and on all impacted services, please check [[phab:T289130|on Phabricator]].

A banner will be displayed 30 minutes before the operation.

Please help your community to be aware of this maintenance operation. {{Int:Feedback-thanks-title}}&lt;section end=&quot;MassMessage&quot;/&gt;

20:35, 24 August 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21927201 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/35|Tech News: 2021-35]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/35|Translations]] are available.

'''Recent changes'''
* Some musical score syntax no longer works and may needed to be updated, you can check [[:Category:{{MediaWiki:score-error-category}}]] on your wiki for a list of pages with errors.

'''Problems'''
* Musical scores were unable to render lyrics in some languages because of missing fonts. This has been fixed now. If your language would prefer a different font, please file a request in Phabricator. [https://phabricator.wikimedia.org/T289554]

'''Changes later this week'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]] about this.
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-01|en}}. It will be on all wikis from {{#time:j xg|2021-09-02|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* You will be able to read but not edit [[phab:T289660|Commons]] for a few minutes on {{#time:j xg|2021-09-06|en}}. This will happen around [https://zonestamp.toolforge.org/1630818058 05:00 UTC]. This is for database maintenance.
* All wikis will be read-only for a few minutes in the week of 13 September. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T287539]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:01, 30 August 2021 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21954810 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/36|Tech News: 2021-36]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/36|Translations]] are available.

'''Recent changes'''
* The wikis that have [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] deployed have been part of A/B testing since deployment, in which some newcomers did not receive the new features. Now, all of the newcomers on 21 of the smallest of those wikis will be receiving the features. [https://phabricator.wikimedia.org/T289786]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] In 2017, the provided jQuery library was upgraded from version 1 to 3, with a compatibility layer. The migration will soon finish, to make the site load faster for everyone. If you maintain a gadget or user script, check if you have any JQMIGRATE errors and fix them, or they will break. [https://phabricator.wikimedia.org/T280944][https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/6Z2BVLOBBEC2QP4VV4KOOVQVE52P3HOP/]
* Last year, the Portuguese Wikipedia community embarked on an experiment to make log-in compulsory for editing.  The [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Impact report for Login Required Experiment on Portuguese Wikipedia|impact report of this trial]] is ready. Moving forward, the Anti-Harassment Tools team is looking for projects that are willing to experiment with restricting IP editing on their wiki for a short-term experiment. [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Login Required Experiment|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:20, 6 September 2021 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21981010 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/37|Tech News: 2021-37]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/37|Translations]] are available.

'''Recent changes'''
* 45 new Wikipedias now have access to the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. [https://phabricator.wikimedia.org/T289680]
* [[mw:Special:MyLanguage/Growth/Deployment table|A majority of Wikipedias]] now have access to the Growth features. The Growth team [[mw:Special:MyLanguage/Growth/FAQ|has published an FAQ page]] about the features. This translatable FAQ covers the description of the features, how to use them, how to change the configuration, and more.

'''Problems'''
* [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on 14 September. This is planned at [https://zonestamp.toolforge.org/1631628002 14:00 UTC]. [https://phabricator.wikimedia.org/T287539]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-15|en}}. It will be on all wikis from {{#time:j xg|2021-09-16|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).
* Starting this week, Wikipedia in Italian will receive weekly software updates on Wednesdays. It used to receive the updates on Thursdays. Due to this change, bugs will be noticed and fixed sooner. [https://phabricator.wikimedia.org/T286664]
* You can add language links in the sidebar in [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|the new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206]
* The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlight]] tool marks up code with different colours. It now can highlight 23 new code languages. Additionally, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;golang&lt;/code&gt;&lt;/bdi&gt; can now be used as an alias for the [[d:Q37227|Go programming language]], and a special &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;output&lt;/code&gt;&lt;/bdi&gt; mode has been added to show a program's output. [https://phabricator.wikimedia.org/T280117][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/SyntaxHighlight_GeSHi/+/715277/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:35, 13 September 2021 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22009517 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/38|Tech News: 2021-38]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/38|Translations]] are available.

'''Recent changes'''
* Growth features are now deployed to almost all Wikipedias. [[phab:T290582|For the majority of small Wikipedias]], the features are only available for experienced users, to [[mw:Special:MyLanguage/Growth/FAQ#enable|test the features]] and [[mw:Special:MyLanguage/Growth/FAQ#config|configure them]]. Features will be available for newcomers starting on 20 September 2021.
* MediaWiki had a feature that would highlight local links to short articles in a different style. Each user could pick the size at which &quot;stubs&quot; would be highlighted. This feature was very bad for performance, and following a consultation, has been removed. [https://phabricator.wikimedia.org/T284917]
* A technical change was made to the MonoBook skin to allow for easier maintenance and upkeep. This has resulted in some minor changes to HTML that make MonoBook's HTML consistent with other skins. Efforts have been made to minimize the impact on editors, but please ping [[m:User:Jon (WMF)|Jon (WMF)]] on wiki or in [[phab:T290888|phabricator]] if any problems are reported.

'''Problems'''
* There was a problem with search last week. Many search requests did not work for 2 hours because of an accidental restart of the search servers. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-09-13_cirrussearch_restart]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-22|en}}. It will be on all wikis from {{#time:j xg|2021-09-23|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[s:Special:ApiHelp/query+proofreadinfo|meta=proofreadpage API]] has changed. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;piprop&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; parameter has been renamed to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;prpiprop&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. API users should update their code to avoid unrecognized parameter warnings. Pywikibot users should upgrade to 6.6.0. [https://phabricator.wikimedia.org/T290585]

'''Future changes'''
* The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks.  It is currently part of &quot;{{int:discussiontools-preference-label}}&quot; in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [https://phabricator.wikimedia.org/T262331]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:MediaWiki_1.37/Deprecation_of_legacy_API_token_parameters|previously announced]] change to how you obtain tokens from the API has been delayed to September 21 because of an incompatibility with Pywikibot. Bot operators using Pywikibot can follow [[phab:T291202|T291202]] for progress on a fix, and should plan to upgrade to 6.6.1 when it is released.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

18:32, 20 September 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22043415 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/39|Tech News: 2021-39]] ==

&lt;section begin=&quot;technews-2021-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/39|Translations]] are available.

'''Recent changes'''
* [[w:en:IOS|iOS 15]] has a new function called [https://support.apple.com/en-us/HT212614 Private Relay] (Apple website). This can hide the user's IP when they use [[w:en:Safari (software)|Safari]] browser. This is like using a [[w:en:Virtual private network|VPN]] in that we see another IP address instead. It is opt-in and only for those who pay extra for [[w:en:ICloud|iCloud]]. It will come to Safari users on [[:w:en:OSX|OSX]] later. There is a [[phab:T289795|technical discussion]] about what this means for the Wikimedia wikis.

'''Problems'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and user-scripts add items to the [[m:Customization:Explaining_skins#Portlets|portlets]] (article tools) part of the skin. A recent change to the HTML may have made those links a different font-size. This can be fixed by adding the CSS class &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.vector-menu-dropdown-noicon&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T291438]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-29|en}}. It will be on all wikis from {{#time:j xg|2021-09-30|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* The [[mw:Special:MyLanguage/Onboarding_new_Wikipedians#New_experience|GettingStarted extension]] was built in 2013, and provides an onboarding process for new account holders in a few versions of Wikipedia. However, the recently developed [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] provide a better onboarding experience. Since the vast majority of Wikipedias now have access to the Growth features, GettingStarted will be deactivated starting on 4 October. [https://phabricator.wikimedia.org/T235752]
* A small number of users will not be able to connect to the Wikimedia wikis after 30 September. This is because an old [[:w:en:root certificate|root certificate]] will no longer work. They will also have problems with many other websites. Users who have updated their software in the last five years are unlikely to have problems. Users in Europe, Africa and Asia are less likely to have immediate problems even if their software is too old. You can [[m:Special:MyLanguage/HTTPS/2021 Let's Encrypt root expiry|read more]].
* You can [[mw:Special:MyLanguage/Help:Notifications|receive notifications]] when someone leaves a comment on user talk page or mentions you in a talk page comment. Clicking the notification link will now bring you to the comment and highlight it. Previously, doing so brought you to the top of the section that contained the comment. You can find [[phab:T282029|more information in T282029.]]

'''Future changes'''
* The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks. It is currently part of &quot;{{int:discussiontools-preference-label}}&quot; in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [[phab:T288485|See the list of wikis.]] [https://phabricator.wikimedia.org/T262331]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W39&quot;/&gt;

22:23, 27 September 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22077885 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/40|Tech News: 2021-40]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/40|Translations]] are available.

'''Recent changes'''
* A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them has been enabled for the following 10 wikis: mediawiki.org, the Italian, Catalan, Hebrew and Vietnamese Wikipedias, French Wikisource, and English Wikivoygage, Wikibooks, Wiktionary and Wikinews. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist on those wikis you can [[phab:T48643|let the developers know]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-06|en}}. It will be on all wikis from {{#time:j xg|2021-10-07|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and bots that use the API to read the AbuseFilter log might break. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;hidden&lt;/code&gt;&lt;/bdi&gt; property will no longer say an entry is &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;implicit&lt;/code&gt;&lt;/bdi&gt; for unsuppressed log entries about suppressed edits. If your bot needs to know this, do a separate revision query. Additionally, the property will have the value &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;false&lt;/code&gt;&lt;/bdi&gt; for visible entries; previously, it wasn't included in the response. [https://phabricator.wikimedia.org/T291718]
* A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them will be enabled for ''all production wikis''. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist you can [[phab:T48643|let the developers know]].

'''Future changes'''
* You can soon get cross-wiki notifications in the [[mw:Wikimedia Apps/Team/iOS|iOS Wikipedia app]]. You can also get notifications as push notifications. More notification updates will follow in later versions. [https://www.mediawiki.org/wiki/Wikimedia_Apps/Team/iOS/Notifications#September_2021_update]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgExtraSignatureNamespaces&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgLegalTitleChars&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgIllegalFileChars&lt;/code&gt;&lt;/bdi&gt; will soon be removed from &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]&lt;/code&gt;&lt;/bdi&gt;. These are not part of the &quot;stable&quot; variables available for use in wiki JavaScript. [https://phabricator.wikimedia.org/T292011] 
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookiePrefix&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookieDomain&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookiePath&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookieExpiration&lt;/code&gt;&lt;/bdi&gt; will soon be removed from mw.config. Scripts should instead use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw.cookie&lt;/code&gt;&lt;/bdi&gt; from the &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[mw:ResourceLoader/Core_modules#mediawiki.cookie|mediawiki.cookie]]&lt;/bdi&gt;&quot; module. [https://phabricator.wikimedia.org/T291760]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:32, 4 October 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22101208 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/41|Tech News: 2021-41]] ==

&lt;section begin=&quot;technews-2021-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/41|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-13|en}}. It will be on all wikis from {{#time:j xg|2021-10-14|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

* The [[mw:Manual:Table_of_contents#Auto-numbering|&quot;auto-number headings&quot; preference]] is being removed. You can read [[phab:T284921]] for the reasons and discussion. This change was [[m:Tech/News/2021/26|previously]] announced. [[mw:Snippets/Auto-number_headings|A JavaScript snippet]] is available which can be used to create a Gadget on wikis that still want to support auto-numbering.

'''Meetings'''
* You can join a meeting about the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. A demonstration version of the [[mw:Reading/Web/Desktop Improvements/Features/Sticky Header|newest feature]] will be shown. The event will take place on Tuesday, 12 October at 16:00 UTC. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web/12-10-2021|See how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W41&quot;/&gt;

15:30, 11 October 2021 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22152137 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/42|Tech News: 2021-42]] ==

&lt;section begin=&quot;technews-2021-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/42|Translations]] are available.

'''Recent changes'''
*[[m:Toolhub|Toolhub]] is a catalogue to make it easier to find software tools that can be used for working on the Wikimedia projects. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LF4SSR4QRCKV6NPRFGUAQWUFQISVIPTS/ read more].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-20|en}}. It will be on all wikis from {{#time:j xg|2021-10-21|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* The developers of the [[mw:Wikimedia Apps/Team/Android|Wikipedia Android app]] are working on [[mw:Wikimedia Apps/Team/Android/Communication|communication in the app]]. You can now answer questions in [[mw:Wikimedia Apps/Team/Android/Communication/UsertestingOctober2021|survey]] to help the development.
* 3–5% of editors may be blocked in the next few months. This is because of a new service in Safari, which is similar to a [[w:en:Proxy server|proxy]] or a [[w:en:VPN|VPN]]. It is called iCloud Private Relay. There is a [[m:Special:MyLanguage/Apple iCloud Private Relay|discussion about this]] on Meta. The goal is to learn what iCloud Private Relay could mean for the communities.
* [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] is a new [[w:en:API|API]] for those who use a lot of information from the Wikimedia projects on other sites. It is a way to get big commercial users to pay for the data. There will soon be a copy of the Wikimedia Enterprise dataset. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/message/B2AX6PWH5MBKB4L63NFZY3ADBQG7MSBA/ read more]. You can also ask the team questions [https://wikimedia.zoom.us/j/88994018553 on Zoom] on [https://www.timeanddate.com/worldclock/fixedtime.html?hour=15&amp;min=00&amp;sec=0&amp;day=22&amp;month=10&amp;year=2021 22 October 15:00 UTC].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W42&quot;/&gt;

20:53, 18 October 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22176877 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/43|Tech News: 2021-43]] ==

&lt;section begin=&quot;technews-2021-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/43|Translations]] are available.

'''Recent changes'''
* The [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award 2021]] is looking for nominations. You can recommend tools until 27 October.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-27|en}}. It will be on all wikis from {{#time:j xg|2021-10-28|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
*[[m:Special:MyLanguage/Help:Diff|Diff pages]] will have an improved copy and pasting experience. [[m:Special:MyLanguage/Community Wishlist Survey 2021/Copy paste diffs|The changes]] will allow the text in the diff for before and after to be treated as separate columns and will remove any unwanted syntax. [https://phabricator.wikimedia.org/T192526]
* The version of the [[w:en:Liberation fonts|Liberation fonts]] used in SVG files will be upgraded. Only new thumbnails will be affected. Liberation Sans Narrow will not change. [https://phabricator.wikimedia.org/T253600]

'''Meetings'''
* You can join a meeting about the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. News about the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Warn when linking to disambiguation pages|disambiguation]] and the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Real Time Preview for Wikitext|real-time preview]] wishes will be shown. The event will take place on Wednesday, 27 October at 14:30 UTC. [[m:Special:MyLanguage/Community Wishlist Survey/Updates/Talk to Us|See how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W43&quot;/&gt;

20:08, 25 October 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22232718 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/44|Tech News: 2021-44]] ==

&lt;section begin=&quot;technews-2021-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/44|Translations]] are available.

'''Recent changes'''
* There is a limit on the amount of emails a user can send each day. This limit is now global instead of per-wiki. This change is to prevent abuse. [https://phabricator.wikimedia.org/T293866]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-03|en}}. It will be on all wikis from {{#time:j xg|2021-11-04|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W44&quot;/&gt;

20:28, 1 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22269406 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/45|Tech News: 2021-45]] ==

&lt;section begin=&quot;technews-2021-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/45|Translations]] are available.

'''Recent changes'''
* Mobile IP editors are now able to receive warning notices indicating they have a talk page message on the mobile website (similar to the orange banners available on desktop). These notices will be displayed on every page outside of the main namespace and every time the user attempts to edit. The notice on desktop now has a slightly different colour. [https://phabricator.wikimedia.org/T284642][https://phabricator.wikimedia.org/T278105]

'''Changes later this week'''
* [[phab:T294321|Wikidata will be read-only]] for a few minutes on 11 November. This will happen around [https://zonestamp.toolforge.org/1636610400 06:00 UTC]. This is for database maintenance. [https://phabricator.wikimedia.org/T294321]
* There is no new MediaWiki version this week.

'''Future changes'''
* In the future, unregistered editors will be given an identity that is not their [[:w:en:IP address|IP address]]. This is for legal reasons. A new user right will let editors who need to know the IPs of unregistered accounts to fight vandalism, spam, and harassment, see the IP. You can read the [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#IP Masking Implementation Approaches (FAQ)|suggestions for how that identity could work]] and [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation|discuss on the talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W45&quot;/&gt;

20:36, 8 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22311003 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/46|Tech News: 2021-46]] ==

&lt;section begin=&quot;technews-2021-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/46|Translations]] are available.

'''Recent changes'''
* Most [[c:Special:MyLanguage/Commons:Maximum_file_size#MAXTHUMB|large file uploads]] errors that had messages like &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;stashfailed&lt;/code&gt;&lt;/bdi&gt;&quot; or &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;DBQueryError&lt;/code&gt;&lt;/bdi&gt;&quot; have now been fixed. An [[wikitech:Incident documentation/2021-11-04 large file upload timeouts|incident report]] is available.

'''Problems'''
* Sometimes, edits made on iOS using the visual editor save groups of numbers as telephone number links, because of a feature in the operating system. This problem is under investigation. [https://phabricator.wikimedia.org/T116525]
* There was a problem with search last week. Many search requests did not work for 2 hours because of a configuration error. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-11-10_cirrussearch_commonsfile_outage]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-17|en}}. It will be on all wikis from {{#time:j xg|2021-11-18|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W46&quot;/&gt;

22:06, 15 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22338097 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/47|Tech News: 2021-47]] ==

&lt;section begin=&quot;technews-2021-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/47|Translations]] are available.

'''Changes later this week'''
* There is no new MediaWiki version this week.
*The template dialog in VisualEditor and in the [[Special:Preferences#mw-prefsection-betafeatures|new wikitext mode]] Beta feature will be [[m:WMDE Technical Wishes/VisualEditor template dialog improvements|heavily improved]] on [[phab:T286992|a few wikis]]. Your [[m:Talk:WMDE Technical Wishes/VisualEditor template dialog improvements|feedback is welcome]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W47&quot;/&gt;

20:02, 22 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22366010 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/48|Tech News: 2021-48]] ==

&lt;section begin=&quot;technews-2021-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/48|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-01|en}}. It will be on all wikis from {{#time:j xg|2021-12-02|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W48&quot;/&gt;

21:15, 29 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22375666 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/49|Tech News: 2021-49]] ==

&lt;section begin=&quot;technews-2021-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/49|Translations]] are available.

'''Problems'''
* MediaWiki 1.38-wmf.11 was scheduled to be deployed on some wikis last week. The deployment was delayed because of unexpected problems.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-08|en}}. It will be on all wikis from {{#time:j xg|2021-12-09|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* At all Wikipedias, a Mentor Dashboard is now available at &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Special:MentorDashboard&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. It allows registered mentors, who take care of newcomers' first steps, to monitor their assigned newcomers' activity. It is part of the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. You can learn more about [[mw:Special:MyLanguage/Growth/Communities/How_to_configure_the_mentors%27_list|activating the mentor list]] on your wiki and about [[mw:Special:MyLanguage/Growth/Mentor dashboard|the mentor dashboard project]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The predecessor to the current [[mw:API|MediaWiki Action API]] (which was created in 2008), &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=ajax&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, will be removed this week. Any scripts or bots using it will need to switch to the corresponding API module. [https://phabricator.wikimedia.org/T42786]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] An old ResourceLoader module, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;jquery.jStorage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, which was deprecated in 2016, will be removed this week. Any scripts or bots using it will need to switch to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mediawiki.storage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; instead. [https://phabricator.wikimedia.org/T143034]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W49&quot;/&gt;

21:59, 6 December 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22413926 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/50|Tech News: 2021-50]] ==

&lt;section begin=&quot;technews-2021-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/50|Translations]] are available.

'''Recent changes'''
* There are now default [[m:Special:MyLanguage/Help:Namespace#Other_namespace_aliases|short aliases]] for the &quot;Project:&quot; namespace on most wikis. E.g. On Wikibooks wikis, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;[[WB:]]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will go to the local language default for the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;[[Project:]]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; namespace. This change is intended to help the smaller communities have easy access to this feature. Additional local aliases can still be requested via [[m:Special:MyLanguage/Requesting wiki configuration changes|the usual process]]. [https://phabricator.wikimedia.org/T293839]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-15|en}}. It will be on all wikis from {{#time:j xg|2021-12-16|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W50&quot;/&gt;

22:27, 13 December 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22441074 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/51|Tech News: 2021-51]] ==

&lt;section begin=&quot;technews-2021-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/51|Translations]] are available.

'''Tech News'''
* Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 10 January 2022.

'''Recent changes'''
* Queries made by the DynamicPageList extension (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;DynamicPageList&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;) are now only allowed to run for 10 seconds and error if they take longer. This is in response to multiple outages where long-running queries caused an outage on all wikis. [https://phabricator.wikimedia.org/T287380#7575719]

'''Changes later this week'''
* There is no new MediaWiki version this week or next week.

'''Future changes'''
* The developers of the Wikipedia iOS app are looking for testers who edit in multiple languages. You can [[mw:Wikimedia Apps/Team/iOS/202112 testing|read more and let them know if you are interested]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The Wikimedia [[wikitech:Portal:Cloud VPS|Cloud VPS]] hosts technical projects for the Wikimedia movement. Developers need to [[wikitech:News/Cloud VPS 2021 Purge|claim projects]] they use. This is because old and unused projects are removed once a year. Unclaimed projects can be shut down from February. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/2B7KYL5VLQNHGQQHMYLW7KTUKXKAYY3T/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W51&quot;/&gt;

22:05, 20 December 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22465395 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/02|Tech News: 2022-02]] ==

&lt;section begin=&quot;technews-2022-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/02|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] A &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;oauth_consumer&lt;/code&gt;&lt;/bdi&gt; variable has been added to the [[mw:Special:MyLanguage/AbuseFilter|AbuseFilter]] to enable identifying changes made by specific tools. [https://phabricator.wikimedia.org/T298281]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets are [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#Package_Gadgets|now able to directly include JSON pages]]. This means some gadgets can now be configured by administrators without needing the interface administrator permission, such as with the Geonotice gadget. [https://phabricator.wikimedia.org/T198758]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets [[mw:Extension:Gadgets#Options|can now specify page actions]] on which they are available. For example, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|actions=edit,history&lt;/code&gt;&lt;/bdi&gt; will load a gadget only while editing and on history pages. [https://phabricator.wikimedia.org/T63007]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets can now be loaded on demand with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;withgadget&lt;/code&gt;&lt;/bdi&gt; URL parameter. This can be used to replace [[mw:Special:MyLanguage/Snippets/Load JS and CSS by URL|an earlier snippet]] that typically looks like &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;withJS&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;withCSS&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T29766]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] At wikis where [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|the Mentorship system is configured]], you can now use the Action API to get a list of a [[mw:Special:MyLanguage/Growth/Mentor_dashboard|mentor's]] mentees. [https://phabricator.wikimedia.org/T291966]
* The heading on the main page can now be configured using &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Mainpage-title-loggedin]]&lt;/span&gt; for logged-in users and &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Mainpage-title]]&lt;/span&gt; for logged-out users. Any CSS that was previously used to hide the heading should be removed. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Small_wiki_toolkits/Starter_kit/Main_page_customization#hide-heading] [https://phabricator.wikimedia.org/T298715]
* Four special pages (and their API counterparts) now have a maximum database query execution time of 30 seconds. These special pages are: RecentChanges, Watchlist, Contributions, and Log. This change will help with site performance and stability. You can read [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IPJNO75HYAQWIGTHI5LJHTDVLVOC4LJP/ more details about this change] including some possible solutions if this affects your workflows. [https://phabricator.wikimedia.org/T297708]
* The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Sticky Header|sticky header]] has been deployed for 50% of logged-in users on [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Frequently asked questions#pilot-wikis|more than 10 wikis]]. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Participate|how to take part in the project]]. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-12|en}}. It will be on all wikis from {{#time:j xg|2022-01-13|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Events'''
* [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] begins. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W02&quot;/&gt;

01:23, 11 January 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22562156 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/03|Tech News: 2022-03]] ==

&lt;section begin=&quot;technews-2022-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/03|Translations]] are available.

'''Recent changes'''
* When using [[mw:Special:MyLanguage/Extension:WikiEditor|WikiEditor]] (also known as the 2010 wikitext editor), people will now see a warning if they link to disambiguation pages.  If you click &quot;{{int:Disambiguator-review-link}}&quot; in the warning, it will ask you to correct the link to a more specific term. You can [[m:Community Wishlist Survey 2021/Warn when linking to disambiguation pages#Jan 12, 2021: Turning on the changes for all Wikis|read more information]] about this completed 2021 Community Wishlist item.
* You can [[mw:Special:MyLanguage/Help:DiscussionTools#subscribe|automatically subscribe to all of the talk page discussions]] that you start or comment in using [[mw:Special:MyLanguage/Talk pages project/Feature summary|DiscussionTools]].  You will receive [[mw:Special:MyLanguage/Notifications|notifications]] when another editor replies.  This is available at most wikis.  Go to your [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]] and turn on &quot;{{int:discussiontools-preference-autotopicsub}}&quot;. [https://phabricator.wikimedia.org/T263819]
* When asked to create a new page or talk page section, input fields can be [[mw:Special:MyLanguage/Manual:Creating_pages_with_preloaded_text|&quot;preloaded&quot; with some text]]. This feature is now limited to wikitext pages. This is so users can't be tricked into making malicious edits. There is a discussion about [[phab:T297725|if this feature should be re-enabled]] for some content types.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-19|en}}. It will be on all wikis from {{#time:j xg|2022-01-20|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Events'''
* [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] continues. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W03&quot;/&gt;

19:55, 17 January 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22620285 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/04|Tech News: 2022-04]] ==

&lt;section begin=&quot;technews-2022-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/04|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-26|en}}. It will be on all wikis from {{#time:j xg|2022-01-27|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* The following languages can now be used with [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighting]]: BDD, Elpi, LilyPond, Maxima, Rita, Savi, Sed, Sophia, Spice, .SRCINFO.
* You can now access your watchlist from outside of the user menu in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]]. The watchlist link appears next to the notification icons if you are at the top of the page. [https://phabricator.wikimedia.org/T289619]

'''Events'''
* You can see the results of the [[m:Special:MyLanguage/Coolest Tool Award|Coolest Tool Award 2021]] and learn more about 14 tools which were selected this year.
* You can [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translate, promote]], or comment on [[m:Special:MyLanguage/Community Wishlist Survey 2022/Proposals|the proposals]] in the Community Wishlist Survey. Voting will begin on {{#time:j xg|2022-01-28|en}}.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W04&quot;/&gt;

21:38, 24 January 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22644148 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/05|Tech News: 2022-05]] ==

&lt;section begin=&quot;technews-2022-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/05|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] If a gadget should support the new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;?withgadget&lt;/code&gt;&lt;/bdi&gt; URL parameter that was [[m:Special:MyLanguage/Tech/News/2022/02|announced]] 3 weeks ago, then it must now also specify &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;supportsUrlLoad&lt;/code&gt;&lt;/bdi&gt; in the gadget definition ([[mw:Special:MyLanguage/Extension:Gadgets#supportsUrlLoad|documentation]]). [https://phabricator.wikimedia.org/T29766]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-02|en}}. It will be on all wikis from {{#time:j xg|2022-02-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* A change that was [[m:Special:MyLanguage/Tech/News/2021/16|announced]] last year was delayed. It is now ready to move ahead:
** The user group &lt;code&gt;oversight&lt;/code&gt; will be renamed &lt;code&gt;suppress&lt;/code&gt;. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in three weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections. As usual, these labels can be translated on translatewiki ([[phab:T112147|direct links are available]]) or by administrators on your wiki.

'''Events'''
* You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey]] between 28 January and 11 February. The survey decides what the [[m:Special:MyLanguage/Community Tech|Community Tech team]] will work on.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W05&quot;/&gt;

17:42, 31 January 2022 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22721804 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/06|Tech News: 2022-06]] ==

&lt;section begin=&quot;technews-2022-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/06|Translations]] are available.

'''Recent changes'''
* English Wikipedia recently set up a gadget for dark mode. You can enable it there, or request help from an [[m:Special:MyLanguage/Interface administrators|interface administrator]] to set it up on your wiki ([[w:en:Wikipedia:Dark mode (gadget)|instructions and screenshot]]).
* Category counts are sometimes wrong. They will now be completely recounted at the beginning of every month. [https://phabricator.wikimedia.org/T299823]

'''Problems'''
* A code-change last week to fix a bug with [[mw:Special:MyLanguage/Manual:Live preview|Live Preview]] may have caused problems with some local gadgets and user-scripts. Any code with skin-specific behaviour for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;vector&lt;/code&gt;&lt;/bdi&gt; should be updated to also check for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;vector-2022&lt;/code&gt;&lt;/bdi&gt;. [[phab:T300987|A code-snippet, global search, and example are available]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-09|en}}. It will be on all wikis from {{#time:j xg|2022-02-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W06&quot;/&gt;

21:15, 7 February 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22765948 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/07|Tech News: 2022-07]] ==

&lt;section begin=&quot;technews-2022-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/07|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/Manual:Purge|Purging]] a category page with fewer than 5,000 members will now recount it completely. This will allow editors to fix incorrect counts when it is wrong. [https://phabricator.wikimedia.org/T85696]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-16|en}}. It will be on all wikis from {{#time:j xg|2022-02-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, the &lt;code dir=ltr&gt;rmspecials()&lt;/code&gt; function has been updated so that it does not remove the &quot;space&quot; character. Wikis are advised to wrap all the uses of &lt;code dir=ltr&gt;rmspecials()&lt;/code&gt; with &lt;code dir=ltr&gt;rmwhitespace()&lt;/code&gt; wherever necessary to keep filters' behavior unchanged. You can use the search function on [[Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T263024]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W07&quot;/&gt;

19:18, 14 February 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22821788 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/08|Tech News: 2022-08]] ==

&lt;section begin=&quot;technews-2022-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/08|Translations]] are available.

'''Recent changes'''
* [[Special:Nuke|Special:Nuke]] will now provide the standard deletion reasons (editable at &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Deletereason-dropdown]]&lt;/bdi&gt;) to use when mass-deleting pages. This was [[m:Community Wishlist Survey 2022/Admins and patrollers/Mass-delete to offer drop-down of standard reasons, or templated reasons.|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T25020]
* At Wikipedias, all new accounts now get the [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] by default when creating an account. Communities are encouraged to [[mw:Special:MyLanguage/Help:Growth/Tools/Account_creation|update their help resources]]. Previously, only 80% of new accounts would get the Growth features. A few Wikipedias remain unaffected by this change. [https://phabricator.wikimedia.org/T301820]
* You can now prevent specific images that are used in a page from appearing in other locations, such as within PagePreviews or Search results. This is done with the markup &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;class=notpageimage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. For example, &lt;code&gt;&lt;nowiki&gt;[[File:Example.png|class=notpageimage]]&lt;/nowiki&gt;&lt;/code&gt;. [https://phabricator.wikimedia.org/T301588]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] There has been a change to the HTML of Special:Contributions, Special:MergeHistory, and History pages, to support the grouping of changes by date in [[mw:Special:MyLanguage/Skin:Minerva_Neue|the mobile skin]]. While unlikely, this may affect gadgets and user scripts. A [[phab:T298638|list of all the HTML changes]] is on Phabricator.

'''Events'''
* [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey results]] have been published. The [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2022 results#leaderboard|ranking of prioritized proposals]] is also available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-23|en}}. It will be on all wikis from {{#time:j xg|2022-02-24|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* The software to play videos and audio files on pages will change soon on all wikis. The old player will be removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Toolforge's underlying operating system is being updated. If you maintain any tools there, there are two options for migrating your tools into the new system.  There are [[wikitech:News/Toolforge Stretch deprecation|details, deadlines, and instructions]] on Wikitech. [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EPJFISC52T7OOEFH5YYMZNL57O4VGSPR/]
* Administrators will soon have [[m:Special:MyLanguage/Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete]] the associated &quot;talk&quot; page when they are deleting a given page. An API endpoint with this option will also be available. This was [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|a request from the 2021 Wishlist Survey]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W08&quot;/&gt;

19:12, 21 February 2022 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22847768 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/09|Tech News: 2022-09]] ==

&lt;section begin=&quot;technews-2022-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/09|Translations]] are available.

'''Recent changes'''
* When searching for edits by [[mw:Special:MyLanguage/Help:Tags|change tags]], e.g. in page history or user contributions, there is now a dropdown list of possible tags. This was [[m:Community Wishlist Survey 2022/Miscellaneous/Improve plain-text change tag selector|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T27909]
* Mentors using the [[mw:Special:MyLanguage/Growth/Mentor_dashboard|Growth Mentor dashboard]] will now see newcomers assigned to them who have made at least one edit, up to 200 edits. Previously, all newcomers assigned to the mentor were visible on the dashboard, even ones without any edit or ones who made hundred of edits. Mentors can still change these values using the filters on their dashboard. Also, the last choice of filters will now be saved. [https://phabricator.wikimedia.org/T301268][https://phabricator.wikimedia.org/T294460]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The user group &lt;code&gt;oversight&lt;/code&gt; was renamed &lt;code&gt;suppress&lt;/code&gt;. This is for [[phab:T109327|technical reasons]]. You may need to update any local references to the old name, e.g. gadgets, links to Special:Listusers, or uses of [[mw:Special:MyLanguage/Help:Magic_words|NUMBERINGROUP]].

'''Problems'''
* The recent change to the HTML of [[mw:Special:MyLanguage/Help:Tracking changes|tracking changes]] pages caused some problems for screenreaders. This is being fixed. [https://phabricator.wikimedia.org/T298638]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-02|en}}. It will be on all wikis from {{#time:j xg|2022-03-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* Working with templates will become easier. [[m:WMDE_Technical_Wishes/Templates|Several improvements]] are planned for March 9 on most wikis and on March 16 on English Wikipedia. The improvements include: Bracket matching, syntax highlighting colors, finding and inserting templates, and related visual editor features.
* If you are a template developer or an interface administrator, and you are intentionally overriding or using the default CSS styles of user feedback boxes (the classes: &lt;code dir=ltr&gt;successbox, messagebox, errorbox, warningbox&lt;/code&gt;), please note that these classes and associated CSS will soon be removed from MediaWiki core. This is to prevent problems when the same class-names are also used on a wiki. Please let us know by commenting at [[phab:T300314]] if you think you might be affected.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W09&quot;/&gt;

22:59, 28 February 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22902593 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/10|Tech News: 2022-10]] ==

&lt;section begin=&quot;technews-2022-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/10|Translations]] are available.

'''Problems'''
* There was a problem with some interface labels last week. It will be fixed this week. This change was part of ongoing work to simplify the support for skins which do not have active maintainers. [https://phabricator.wikimedia.org/T301203]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-09|en}}. It will be on all wikis from {{#time:j xg|2022-03-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W10&quot;/&gt;

21:16, 7 March 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22958074 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/11|Tech News: 2022-11]] ==

&lt;section begin=&quot;technews-2022-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/11|Translations]] are available.

'''Recent changes'''
* In the Wikipedia Android app [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Android/Communication#Updates|it is now possible]] to change the toolbar at the bottom so the tools you use more often are easier to click on. The app now also has a focused reading mode. [https://phabricator.wikimedia.org/T296753][https://phabricator.wikimedia.org/T254771]

'''Problems'''
* There was a problem with the collection of some page-view data from June 2021 to January 2022 on all wikis. This means the statistics are incomplete. To help calculate which projects and regions were most affected, relevant datasets are being retained for 30 extra days. You can [[m:Talk:Data_retention_guidelines#Added_exception_for_page_views_investigation|read more on Meta-wiki]].
* There was a problem with the databases on March 10. All wikis were unreachable for logged-in users for 12 minutes. Logged-out users could read pages but could not edit or access uncached content then. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2022-03-10_MediaWiki_availability]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-16|en}}. It will be on all wikis from {{#time:j xg|2022-03-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* When [[mw:Special:MyLanguage/Help:System_message#Finding_messages_and_documentation|using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;uselang=qqx&lt;/code&gt;&lt;/bdi&gt; to find localisation messages]], it will now show all possible message keys for navigation tabs such as &quot;{{int:vector-view-history}}&quot;. [https://phabricator.wikimedia.org/T300069]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Access to [[{{#special:RevisionDelete}}]] has been expanded to include users who have &lt;code dir=ltr&gt;deletelogentry&lt;/code&gt; and &lt;code dir=ltr&gt;deletedhistory&lt;/code&gt; rights through their group memberships. Before, only those with the &lt;code dir=ltr&gt;deleterevision&lt;/code&gt; right could access this special page. [https://phabricator.wikimedia.org/T301928]
* On the [[{{#special:Undelete}}]] pages for diffs and revisions, there will be a link back to the main Undelete page with the list of revisions. [https://phabricator.wikimedia.org/T284114]

'''Future changes'''
*  The Wikimedia Foundation has announced the IP Masking implementation strategy and next steps. The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation#feb25|announcement can be read here]].
* The [[mw:Special:MyLanguage/Wikimedia Apps/Android FAQ|Wikipedia Android app]] developers are working on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|new functions]] for user talk pages and article talk pages. [https://phabricator.wikimedia.org/T297617]

'''Events'''
* The [[mw:Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place as a hybrid event on 20-22 May 2022. The Hackathon will be held online and there are grants available to support local in-person meetups around the world. Grants can be requested until 20 March.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W11&quot;/&gt;

22:07, 14 March 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22993074 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/12|Tech News: 2022-12]] ==

&lt;section begin=&quot;technews-2022-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/12|Translations]] are available.

'''New code release schedule for this week'''
* There will be four MediaWiki releases this week, instead of just one. This is an experiment which should lead to fewer problems and to faster feature updates. The releases will be on all wikis, at different times, on Monday, Tuesday, and Wednesday. You can [[mw:Special:MyLanguage/Wikimedia Release Engineering Team/Trainsperiment week|read more about this project]].

'''Recent changes'''
* You can now set how many search results to show by default in [[Special:Preferences#mw-prefsection-searchoptions|your Preferences]]. This was the 12th most popular wish in the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey 2022]]. [https://phabricator.wikimedia.org/T215716]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Jupyter notebooks tool [[wikitech:PAWS|PAWS]] has been updated to a new interface. [https://phabricator.wikimedia.org/T295043]

'''Future changes'''
* Interactive maps via [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] will soon work on wikis using the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions]] extension. [https://wikimedia.sslsurvey.de/Kartographer-Workflows-EN/ Please tell us] which improvements you want to see in Kartographer. You can take this survey in simple English. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W12&quot;/&gt;

16:01, 21 March 2022 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23034693 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/13|Tech News: 2022-13]] ==

&lt;section begin=&quot;technews-2022-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/13|Translations]] are available.

'''Recent changes'''
* There is a simple new Wikimedia Commons upload tool available for macOS users, [[c:Commons:Sunflower|Sunflower]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-30|en}}. It will be on all wikis from {{#time:j xg|2022-03-31|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* Some wikis will be in read-only for a few minutes because of regular database maintenance. It will be performed on {{#time:j xg|2022-03-29|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-03-31|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). [https://phabricator.wikimedia.org/T301850][https://phabricator.wikimedia.org/T303798]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W13&quot;/&gt;

19:54, 28 March 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23073711 --&gt;

== [[m:Special:MyLanguage/Tech/News/2022/14|Tech News: 2022-14]] ==

&lt;section begin=&quot;technews-2022-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/14|Translations]] are available.

'''Problems'''
* For a few days last week, edits that were suggested to newcomers were not tagged in the [[{{#special:recentchanges}}]] feed. This bug has been fixed. [https://phabricator.wikimedia.org/T304747]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-06|en}}. It will be on all wikis from {{#time:j xg|2022-04-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).

'''Future changes'''
* Starting next week, Tech News' title will be translatable. When the newsletter is distributed, its title may not be &lt;code dir=ltr&gt;Tech News: 2022-14&lt;/code&gt; anymore. It may affect some filters that have been set up by some communities. [https://phabricator.wikimedia.org/T302920]
* Over the next few months, the &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; Growth feature [[phab:T304110|will become available to more Wikipedias]]. Each week, a few wikis will get the feature. You can test this tool at [[mw:Special:MyLanguage/Growth#deploymentstable|a few wikis where &quot;Link recommendation&quot; is already available]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W14&quot;/&gt;

21:01, 4 April 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23097604 --&gt;

== Tech News: 2022-15 ==

&lt;section begin=&quot;technews-2022-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/15|Translations]] are available.

'''Recent changes'''
* There is a new public status page at &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikimediastatus.net/ www.wikimediastatus.net]&lt;/span&gt;. This site shows five automated high-level metrics where you can see the overall health and performance of our wikis' technical environment. It also contains manually-written updates for widespread incidents, which are written as quickly as the engineers are able to do so while also fixing the actual problem. The site is separated from our production infrastructure and hosted by an external service, so that it can be accessed even if the wikis are briefly unavailable. You can [https://diff.wikimedia.org/2022/03/31/announcing-www-wikimediastatus-net/ read more about this project].
* On Wiktionary wikis, the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-13|en}}. It will be on all wikis from {{#time:j xg|2022-04-14|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W15&quot;/&gt;

19:44, 11 April 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23124108 --&gt;

== Tech News: 2022-16 ==

&lt;section begin=&quot;technews-2022-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/16|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-20|en}}. It will be on all wikis from {{#time:j xg|2022-04-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-19|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]) and on {{#time:j xg|2022-04-21|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]).
* Administrators will now have [[m:Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete the associated &quot;Talk&quot; page]] when they are deleting a given page. An API endpoint with this option is also available. This concludes the [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|11th wish of the 2021 Community Wishlist Survey]].
* On [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#test-wikis|selected wikis]], 50% of logged-in users will see the new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Table of contents|table of contents]]. When scrolling up and down the page, the table of contents will stay in the same place on the screen. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]] project. [https://phabricator.wikimedia.org/T304169]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Message boxes produced by MediaWiki code will no longer have these CSS classes: &lt;code dir=ltr&gt;successbox&lt;/code&gt;, &lt;code dir=ltr&gt;errorbox&lt;/code&gt;, &lt;code dir=ltr&gt;warningbox&lt;/code&gt;. The styles for those classes and &lt;code dir=ltr&gt;messagebox&lt;/code&gt; will be removed from MediaWiki core. This only affects wikis that use these classes in wikitext, or change their appearance within site-wide CSS. Please review any local usage and definitions for these classes you may have. This was previously announced in the [[m:Special:MyLanguage/Tech/News/2022/09|28 February issue of Tech News]].

'''Future changes'''
* [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] will become compatible with [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions page stabilization]]. Kartographer maps will also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions] The Kartographer documentation has been thoroughly updated. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer/Getting_started] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:VisualEditor/Maps] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W16&quot;/&gt;

23:11, 18 April 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23167004 --&gt;

== Tech News: 2022-17 ==

&lt;section begin=&quot;technews-2022-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/17|Translations]] are available.

'''Recent changes'''
*  On [https://noc.wikimedia.org/conf/dblists/group1.dblist many wikis] (group 1), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-27|en}}. It will be on all wikis from {{#time:j xg|2022-04-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-26|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]).
* Some very old browsers and operating systems are no longer supported. Some things on the wikis might look weird or not work in very old browsers like Internet Explorer 9 or 10, Android 4, or Firefox 38 or older. [https://phabricator.wikimedia.org/T306486]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W17&quot;/&gt;

22:56, 25 April 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23187115 --&gt;

== Tech News: 2022-18 ==

&lt;section begin=&quot;technews-2022-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/18|Translations]] are available.

'''Recent changes'''
* On [https://noc.wikimedia.org/conf/dblists/group2.dblist all remaining wikis] (group 2), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-04|en}}. It will be on all wikis from {{#time:j xg|2022-05-05|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* The developers are working on talk pages in the [[mw:Wikimedia Apps/Team/iOS|Wikipedia app for iOS]]. You can [https://wikimedia.qualtrics.com/jfe/form/SV_9GBcHczQGLbQWTY give feedback]. You can take the survey in English, German, Hebrew or Chinese.
* [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements#Status_and_next_steps|Most wikis]] will receive an [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements|improved template dialog]] in VisualEditor and New Wikitext mode. [https://phabricator.wikimedia.org/T296759] [https://phabricator.wikimedia.org/T306967]
* If you use syntax highlighting while editing wikitext, you can soon activate a [[m:WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting#Color-blind_mode|colorblind-friendly color scheme]]. [https://phabricator.wikimedia.org/T306867]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Several CSS IDs related to MediaWiki interface messages will be removed. Technical editors should please [[phab:T304363|review the list of IDs and links to their existing uses]]. These include &lt;code dir=ltr&gt;#mw-anon-edit-warning&lt;/code&gt;, &lt;code dir=ltr&gt;#mw-undelete-revision&lt;/code&gt; and 3 others.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W18&quot;/&gt;

19:33, 2 May 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23232924 --&gt;

== Tech News: 2022-19 ==

&lt;section begin=&quot;technews-2022-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/19|Translations]] are available.

'''Recent changes'''
* You can now see categories in the [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia app for Android]]. [https://phabricator.wikimedia.org/T73966]

'''Problems'''
* Last week, there was a problem with Wikidata's search autocomplete. This has now been fixed. [https://phabricator.wikimedia.org/T307586]
* Last week, all wikis had slow access or no access for 20 minutes, for logged-in users and non-cached pages. This was caused by a problem with a database change. [https://phabricator.wikimedia.org/T307647]

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://phabricator.wikimedia.org/T305217#7894966]
*  [[m:WMDE Technical Wishes/Geoinformation#Current issues|Incompatibility issues]] with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] and the [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|FlaggedRevs extension]] will be fixed: Deployment is planned for May 10 on all wikis. Kartographer will then be enabled on the [[phab:T307348|five wikis which have not yet enabled the extension]] on May 24.
* The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] skin will be set as the default on several more wikis, including Arabic and Catalan Wikipedias. Logged-in users will be able to switch back to the old Vector (2010). See the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|latest update]] about Vector (2022).

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place on 17 May. The following meetings are currently planned for: 7 June, 21 June, 5 July, 19 July.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W19&quot;/&gt;

15:22, 9 May 2022 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23256717 --&gt;

== Tech News: 2022-20 ==

&lt;section begin=&quot;technews-2022-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/20|Translations]] are available.

'''Changes later this week'''
* Some wikis can soon use the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|add a link]] feature. This will start on Wednesday. The wikis are {{int:project-localized-name-cawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-simplewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-svwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}. This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542]
* The [[mw:Special:MyLanguage/Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place online on May 20–22. It will be in English. There are also local [[mw:Special:MyLanguage/Wikimedia Hackathon 2022/Meetups|hackathon meetups]] in Germany, Ghana, Greece, India, Nigeria and the United States. Technically interested Wikimedians can work on software projects and learn new skills. You can also host a session or post a project you want to work on.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-18|en}}. It will be on all wikis from {{#time:j xg|2022-05-19|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* You can soon edit translatable pages in the visual editor. Translatable pages exist on for examples Meta and Commons. [https://diff.wikimedia.org/2022/05/12/mediawiki-1-38-brings-support-for-editing-translatable-pages-with-the-visual-editor/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W20&quot;/&gt;

18:58, 16 May 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23291515 --&gt;

== Tech News: 2022-21 ==

&lt;section begin=&quot;technews-2022-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/21|Translations]] are available.

'''Recent changes'''
* Administrators using the mobile web interface can now access Special:Block directly from user pages. [https://phabricator.wikimedia.org/T307341]
* The &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wiktionary.org/ www.wiktionary.org]&lt;/span&gt; portal page now uses an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T304629]

'''Problems'''
* The Growth team maintains a mentorship program for newcomers. Previously, newcomers weren't able to opt out from the program. Starting May 19, 2022, newcomers are able to fully opt out from Growth mentorship, in case they do not wish to have any mentor at all. [https://phabricator.wikimedia.org/T287915]
* Some editors cannot access the content translation tool if they load it by clicking from the contributions menu. This problem is being worked on. It should still work properly if accessed directly via Special:ContentTranslation. [https://phabricator.wikimedia.org/T308802]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-25|en}}. It will be on all wikis from {{#time:j xg|2022-05-26|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadget and user scripts developers are invited to give feedback on a [[mw:User:Jdlrobson/Extension:Gadget/Policy|proposed technical policy]] aiming to improve support from MediaWiki developers. [https://phabricator.wikimedia.org/T308686]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W21&quot;/&gt;

00:21, 24 May 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23317250 --&gt;

== Tech News: 2022-22 ==

&lt;section begin=&quot;technews-2022-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/22|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]]  In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, an &lt;code dir=ltr&gt;ip_in_ranges()&lt;/code&gt; function has been introduced to check if an IP is in any of the ranges. Wikis are advised to combine multiple &lt;code dir=ltr&gt;ip_in_range()&lt;/code&gt; expressions joined by &lt;code&gt;|&lt;/code&gt; into a single expression for better performance. You can use the search function on [[Special:AbuseFilter|Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T305017]
* The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature|IP Info feature]] which helps abuse fighters access information about IPs, [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May 24, 2022|has been deployed]] to all wikis as a beta feature. This comes after weeks of beta testing on test.wikipedia.org.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-01|en}}. It will be on all wikis from {{#time:j xg|2022-06-02|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-05-31|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]).
* The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at most wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]]  The [[:mw:Special:ApiHelp/query+usercontribs|list=usercontribs API]] will support fetching contributions from an [[mw:Special:MyLanguage/Help:Range blocks#Non-technical explanation|IP range]] soon. API users can set the &lt;code&gt;uciprange&lt;/code&gt; parameter to get contributions from any IP range within [[:mw:Manual:$wgRangeContributionsCIDRLimit|the limit]]. [https://phabricator.wikimedia.org/T177150]
* A new parser function will be introduced: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{=}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. It will replace existing templates named &quot;=&quot;. It will insert an [[w:en:Equals sign|equal sign]]. This can be used to escape the equal sign in the parameter values of templates. [https://phabricator.wikimedia.org/T91154]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W22&quot;/&gt;

20:28, 30 May 2022 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23340178 --&gt;

== Tech News: 2022-23 ==

&lt;section begin=&quot;technews-2022-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/23|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-08|en}}. It will be on all wikis from {{#time:j xg|2022-06-09|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;str_replace_regexp()&lt;/code&gt;&lt;/bdi&gt; function can be used in [[Special:AbuseFilter|abuse filters]] to replace parts of text using a [[w:en:Regular expression|regular expression]]. [https://phabricator.wikimedia.org/T285468]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W23&quot;/&gt;

02:46, 7 June 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23366979 --&gt;

== Tech News: 2022-24 ==

&lt;section begin=&quot;technews-2022-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/24|Translations]] are available.

'''Recent changes'''
* All wikis can now use [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] maps. Kartographer maps now also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions][https://phabricator.wikimedia.org/T307348]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-15|en}}. It will be on all wikis from {{#time:j xg|2022-06-16|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-14|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]). [https://phabricator.wikimedia.org/T300471]
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-abwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-acewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-adywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-afwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-akwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-alswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-amwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-anwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-angwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-astwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-atjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-avwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-aywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azbwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304548]
* The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at Commons, Wikidata, and some other wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804]

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place today (13 June). The following meetings will take place on: 28 June, 12 July, 26 July.

'''Future changes'''
* By the end of July, the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022]] skin should be ready to become the default across all wikis. Discussions on how to adjust it to the communities' needs will begin in the next weeks. It will always be possible to revert to the previous version on an individual basis. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W24&quot;/&gt;

16:58, 13 June 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23389956 --&gt;

== Tech News: 2022-25 ==

&lt;section begin=&quot;technews-2022-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/25|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia App for Android]] now has an option for editing the whole page at once, located in the overflow menu (three-dots menu [[File:Ic more vert 36px.svg|15px|link=|alt=]]). [https://phabricator.wikimedia.org/T103622]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Some recent database changes may affect queries using the [[m:Research:Quarry|Quarry tool]]. Queries for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;site_stats&lt;/code&gt;&lt;/bdi&gt; at English Wikipedia, Commons, and Wikidata will need to be updated. [[phab:T306589|Read more]].
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;user_global_editcount&lt;/code&gt;&lt;/bdi&gt; variable can be used in [[Special:AbuseFilter|abuse filters]] to avoid affecting globally active users. [https://phabricator.wikimedia.org/T130439]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-22|en}}. It will be on all wikis from {{#time:j xg|2022-06-23|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* Users of non-responsive skins (e.g. MonoBook or Vector) on mobile devices may notice a slight change in the default zoom level. This is intended to optimize zooming and ensure all interface elements are present on the page (for example the table of contents on Vector 2022). In the unlikely event this causes any problems with how you use the site, we'd love to understand better, please ping &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:Jon (WMF)|Jon (WMF)]]&lt;/span&gt; to any on-wiki conversations. [https://phabricator.wikimedia.org/T306910]

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid's HTML output will soon stop annotating file links with different &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;typeof&lt;/code&gt;&lt;/bdi&gt; attribute values, and instead use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:File&lt;/code&gt;&lt;/bdi&gt; for all types. Tool authors should adjust any code that expects: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:Image&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:Audio&lt;/code&gt;&lt;/bdi&gt;, or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:Video&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T273505]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W25&quot;/&gt;

20:18, 20 June 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23425855 --&gt;

== Tech News: 2022-26 ==

&lt;section begin=&quot;technews-2022-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/26|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] API service now has self-service accounts with free on-demand requests and monthly snapshots ([https://enterprise.wikimedia.com/docs/ API documentation]). Community access [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#community-access|via database dumps &amp; Wikimedia Cloud Services]] continues.
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[d:Special:MyLanguage/Wikidata:Wiktionary#lua|All Wikimedia wikis can now use Wikidata Lexemes in Lua]] after creating local modules and templates. Discussions are welcome [[d:Wikidata_talk:Lexicographical_data#You_can_now_reuse_Wikidata_Lexemes_on_all_wikis|on the project talk page]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-29|en}}. It will be on all wikis from {{#time:j xg|2022-06-30|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-28|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T311033]
*  Some global and cross-wiki services will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-30|en}} at 06:00 UTC. This will impact ContentTranslation, Echo, StructuredDiscussions, Growth experiments and a few more services. [https://phabricator.wikimedia.org/T300472]
* Users will be able to sort columns within sortable tables in the mobile skin. [https://phabricator.wikimedia.org/T233340]

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (28 June). The following meetings will take place on 12 July and 26 July.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W26&quot;/&gt;

20:02, 27 June 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23453785 --&gt;

== Tech News: 2022-27 ==

&lt;section begin=&quot;technews-2022-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/27|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-06|en}}. It will be on all wikis from {{#time:j xg|2022-07-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-05|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-07-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].
* [[File:Octicons-tools.svg|15px|link=|alt=| Advanced item]] This change only affects pages in the main namespace in Wikisource. The Javascript config variable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;proofreadpage_source_href&lt;/code&gt;&lt;/bdi&gt; will be removed from &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]&lt;/code&gt;&lt;/bdi&gt; and be replaced with the variable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prpSourceIndexPage&lt;/code&gt;&lt;/bdi&gt;.  [https://phabricator.wikimedia.org/T309490]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W27&quot;/&gt;

19:32, 4 July 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23466250 --&gt;

== Tech News: 2022-28 ==

&lt;section begin=&quot;technews-2022-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/28|Translations]] are available.

'''Recent changes'''
* In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], the page title is now displayed above the tabs such as Discussion, Read, Edit, View history, or More. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates#Page title/tabs switch|Learn more]]. [https://phabricator.wikimedia.org/T303549]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] It is now possible to easily view most of the configuration settings that apply to just one wiki, and to compare settings between two wikis if those settings are different. For example: [https://noc.wikimedia.org/wiki.php?wiki=jawiktionary Japanese Wiktionary settings], or [https://noc.wikimedia.org/wiki.php?wiki=eswiki&amp;compare=eowiki settings that are different between the Spanish and Esperanto Wikipedias]. Local communities may want to [[m:Special:MyLanguage/Requesting_wiki_configuration_changes|discuss and propose changes]] to their local settings. Details about each of the named settings can be found by [[mw:Special:Search|searching MediaWiki.org]]. [https://phabricator.wikimedia.org/T308932]
*The Anti-Harassment Tools team [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May|recently deployed]] the IP Info Feature as a [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature at all wikis]]. This feature allows abuse fighters to access information about IP addresses. Please check our update on [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#April|how to find and use the tool]]. Please share your feedback using a link you will be given within the tool itself.

'''Changes later this week'''
* There is no new MediaWiki version this week.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]).

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W28&quot;/&gt;

19:24, 11 July 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23502519 --&gt;

== Tech News: 2022-29 ==

&lt;section begin=&quot;technews-2022-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/29|Translations]] are available.

'''Problems'''
* The feature on mobile web for [[mw:Special:MyLanguage/Extension:NearbyPages|Nearby Pages]] was missing last week. It will be fixed this week. [https://phabricator.wikimedia.org/T312864]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-20|en}}. It will be on all wikis from {{#time:j xg|2022-07-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* The [[mw:Technical_decision_making/Forum|Technical Decision Forum]] is seeking [[mw:Technical_decision_making/Community_representation|community representatives]]. You can apply on wiki or by emailing &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;TDFSupport@wikimedia.org&lt;/span&gt; before 12 August.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W29&quot;/&gt;

22:59, 18 July 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23517957 --&gt;

== Tech News: 2022-30 ==

&lt;section begin=&quot;technews-2022-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/30|Translations]] are available.

'''Recent changes'''
* The &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikibooks.org/ www.wikibooks.org]&lt;/span&gt; and &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikiquote.org/ www.wikiquote.org]&lt;/span&gt; portal pages now use an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T273179]

'''Problems'''
*  Last week, some wikis were in read-only mode for a few minutes because of an emergency switch of their main database ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T313383]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-27|en}}. It will be on all wikis from {{#time:j xg|2022-07-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* The external link icon will change slightly in the skins Vector legacy and Vector 2022. The new icon uses simpler shapes to be more recognizable on low-fidelity screens. [https://phabricator.wikimedia.org/T261391]
* Administrators will now see buttons on user pages for &quot;{{int:changeblockip}}&quot; and &quot;{{int:unblockip}}&quot; instead of just &quot;{{int:blockip}}&quot; if the user is already blocked. [https://phabricator.wikimedia.org/T308570]

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (26 July).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W30&quot;/&gt;

19:27, 25 July 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23545370 --&gt;

== Tech News: 2022-31 ==

&lt;section begin=&quot;technews-2022-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/31|Translations]] are available.

'''Recent changes'''
* Improved [[m:Special:MyLanguage/Help:Displaying_a_formula#Phantom|LaTeX capabilities for math rendering]] are now available in the wikis thanks to supporting &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Phantom&lt;/code&gt;&lt;/bdi&gt; tags. This completes part of [[m:Community_Wishlist_Survey_2022/Editing/Missing_LaTeX_capabilities_for_math_rendering|the #59 wish]] of the 2022 Community Wishlist Survey.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-03|en}}. It will be on all wikis from {{#time:j xg|2022-08-04|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup0.dblist Group 0]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]].

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].

'''Future meetings'''
* This week, three meetings about [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] with live interpretation will take place. On Tuesday, interpretation in Russian will be provided. On Thursday, meetings for Arabic and Spanish speakers will take place. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|See how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W31&quot;/&gt;

21:21, 1 August 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23615613 --&gt;

== Tech News: 2022-32 ==

&lt;section begin=&quot;technews-2022-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/32|Translations]] are available.

'''Recent changes'''
* [[:m:Special:MyLanguage/Meta:GUS2Wiki/Script|GUS2Wiki]] copies the information from [[{{#special:GadgetUsage}}]] to an on-wiki page so you can review its history. If your project isn't already listed on the [[d:Q113143828|Wikidata entry for Project:GUS2Wiki]] you can either run GUS2Wiki yourself or [[:m:Special:MyLanguage/Meta:GUS2Wiki/Script#Opting|make a request to receive updates]]. [https://phabricator.wikimedia.org/T121049]

'''Changes later this week'''
* There is no new MediaWiki version this week.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-09|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2022-08-11|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]).

'''Future meetings'''
* The [[wmania:Special:MyLanguage/Hackathon|Wikimania Hackathon]] will take place online from August 12–14. Don't miss [[wmania:Special:MyLanguage/Hackathon/Schedule|the pre-hacking showcase]] to learn about projects and find collaborators. Anyone can [[phab:/project/board/6030/|propose a project]] or [[wmania:Special:MyLanguage/Hackathon/Schedule|host a session]]. [[wmania:Special:MyLanguage/Hackathon/Newcomers|Newcomers are welcome]]!

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W32&quot;/&gt;

19:50, 8 August 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23627807 --&gt;

== Tech News: 2022-33 ==

&lt;section begin=&quot;technews-2022-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/33|Translations]] are available.

'''Recent changes'''
* The Persian (Farsi) Wikipedia community decided to block IP editing from October 2021 to April 2022. The Wikimedia Foundation's Product Analytics team tracked the impact of this change. [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Editing Restriction Study/Farsi Wikipedia|An impact report]] is now available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-17|en}}. It will be on all wikis from {{#time:j xg|2022-08-18|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-16|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]) and on {{#time:j xg|2022-08-18|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]).
* The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup1.dblist Group 1]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]].

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. [https://www.mediawiki.org/wiki/Talk_pages_project/Usability#4_August_2022][https://www.mediawiki.org/wiki/Talk_pages_project/Usability#Phase_1:_Topic_containers][https://phabricator.wikimedia.org/T312672]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W33&quot;/&gt;

21:08, 15 August 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23658001 --&gt;

== Tech News: 2022-34 ==

&lt;section begin=&quot;technews-2022-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/34|Translations]] are available.

'''Recent changes'''
* Two problems with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps have been fixed. Maps are no longer shown as empty when a geoline was created via VisualEditor. Geolines consisting of points with QIDs (e.g., subway lines) are no longer shown with pushpins. [https://phabricator.wikimedia.org/T292613][https://phabricator.wikimedia.org/T308560]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-24|en}}. It will be on all wikis from {{#time:j xg|2022-08-25|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-25|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).
* The colours of links and visited links will change. This is to make the difference between links and other text more clear. [https://phabricator.wikimedia.org/T213778]

'''Future changes'''
* The new [{{int:discussiontools-topicsubscription-button-subscribe}}] button [[mw:Talk pages project/Notifications#12 August 2022|helps newcomers get answers]].  The Editing team is enabling this tool everywhere.  You can turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|your preferences]]. [https://phabricator.wikimedia.org/T284489]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W34&quot;/&gt;

00:12, 23 August 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23675501 --&gt;

== Tech News: 2022-35 ==

&lt;section begin=&quot;technews-2022-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/35|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] is available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup2.dblist Group 2]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]]. Please note that when this Beta feature is enabled, it may cause conflicts with some wiki-specific Gadgets.

'''Problems'''
* In recent months, there have been inaccurate numbers shown for various [[{{#special:statistics}}]] at Commons, Wikidata, and English Wikipedia. This has now been fixed. [https://phabricator.wikimedia.org/T315693]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-31|en}}. It will be on all wikis from {{#time:j xg|2022-09-01|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-30|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-09-01|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]).

'''Future changes'''
* The Wikimedia Foundation wants to improve how Wikimedia communities report harmful incidents by building the [[m:Special:MyLanguage/Private Incident Reporting System|Private Incident Reporting System (PIRS)]] to make it easy and safe for users to make reports. You can leave comments on the talk page, by answering the [[m:Special:MyLanguage/Private Incident Reporting System#Phase 1|questions provided]].  If you have ever faced a harmful situation that you wanted to report/reported, join a PIRS interview to share your experience. To sign up [[m:Special:EmailUser/MAna_(WMF)|please email]] &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:MAna (WMF)|Madalina Ana]]&lt;/span&gt;.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W35&quot;/&gt;

23:05, 29 August 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23725814 --&gt;

== Tech News: 2022-36 ==

&lt;section begin=&quot;technews-2022-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/36|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-07|en}}. It will be on all wikis from {{#time:j xg|2022-09-08|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-09-06|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]) and on {{#time:j xg|2022-09-08|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] On Special pages that only have one tab, the tab-bar's row will be hidden in the Vector-2022 skin to save space. The row will still show if Gadgets use it. Gadgets that currently append directly to the CSS id of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;#p-namespaces&lt;/code&gt;&lt;/bdi&gt; should be updated to use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:ResourceLoader/Core_modules#addPortletLink|mw.util.addPortletLink]]&lt;/code&gt;&lt;/bdi&gt; function instead. Gadgets that style this id should consider also targeting &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;#p-associated-pages&lt;/code&gt;&lt;/bdi&gt;, the new id for this row. [[phab:T316908|Examples are available]]. [https://phabricator.wikimedia.org/T316908][https://phabricator.wikimedia.org/T313409]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W36&quot;/&gt;

23:22, 5 September 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23757743 --&gt;

== Tech News: 2022-37 ==

&lt;section begin=&quot;technews-2022-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/37|Translations]] are available.

'''Recent changes'''
* The search servers have been upgraded to a new major version. If you notice any issues with searching, please report them on [[phab:project/view/1849/|Phabricator]]. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/XPCTYYTN67FVFKN6XOHULJVGUO44J662]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-14|en}}. It will be on all wikis from {{#time:j xg|2022-09-15|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[mw:Special:MyLanguage/Extension:SyntaxHighlight|Syntax highlighting]] is now tracked as an [[mw:Special:MyLanguage/Manual:$wgExpensiveParserFunctionLimit|expensive parser function]]. Only 500 expensive function calls can be used on a single page. Pages that exceed the limit are added to a [[:Category:{{MediaWiki:expensive-parserfunction-category}}|tracking category]]. [https://phabricator.wikimedia.org/T316858]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W37&quot;/&gt;

01:50, 13 September 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23787318 --&gt;

== Tech News: 2022-38 ==

&lt;section begin=&quot;technews-2022-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/38|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Two database fields in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;templatelinks&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; table are now being dropped: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tl_namespace&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tl_title&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Any queries that rely on these fields need to be changed to use the new normalization field called &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tl_target_id&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. See &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[phab:T299417|T299417]]&lt;/span&gt; for more information. This is part of [[w:Database normalization|normalization]] of links tables. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/U2U6TXIBABU3KDCVUOITIGI5OJ4COBSW/][https://www.mediawiki.org/wiki/User:ASarabadani_(WMF)/Database_for_devs_toolkit/Concepts/Normalization]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-21|en}}. It will be on all wikis from {{#time:j xg|2022-09-22|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* In [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps, you can use icons on markers for common points of interest. On Tuesday, the [[mw:Special:MyLanguage/Help:Extension:Kartographer/Icons|previous icon set]] will be updated to [https://de.wikipedia.beta.wmflabs.org/wiki/Hilfe:Extension:Kartographer/Icons version maki 7.2]. That means, around 100 new icons will be available.  Additionally, all existing icons were updated for clarity and to make them work better in international contexts. [https://phabricator.wikimedia.org/T302861][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Update_maki_icons]

'''Future changes'''
* In a [[m:Content_Partnerships_Hub/Software/Volunteer_developers_discussion_at_Wikimania_2022|group discussion at Wikimania]], more than 30 people talked about how to make content partnership software in the Wikimedia movement more sustainable. What kind of support is acceptable for volunteer developers? Read the summary and [[m:Talk:Content Partnerships Hub/Software/Volunteer developers discussion at Wikimania 2022|leave your feedback]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W38&quot;/&gt;

&lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/span&gt; 22:16, 19 September 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23826293 --&gt;

== Tech News: 2022-39 ==

&lt;section begin=&quot;technews-2022-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/39|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid clients should be updated to allow for space-separated multi-values in the &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;rel&lt;/code&gt;&lt;/bdi&gt; attribute of links. Further details are in &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[phab:T315209|T315209]]&lt;/bdi&gt;.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-28|en}}. It will be on all wikis from {{#time:j xg|2022-09-29|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[mw:Special:MyLanguage/VisualEditor/Diffs|Visual diffs]] will become available to all users, except at the Wiktionaries and Wikipedias. [https://phabricator.wikimedia.org/T314588]
* [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|Talk pages on the mobile site]] will change at the Arabic, Bangla, Chinese, French, Haitian Creole, Hebrew, Korean, and Vietnamese Wikipedias.  They should be easier to use and provide more information. [https://phabricator.wikimedia.org/T318302] [https://www.mediawiki.org/wiki/Talk_pages_project/Mobile]
* In the [[mw:Lua/Scripting|{{ns:828}}]] namespace, pages ending with &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.json&lt;/code&gt;&lt;/bdi&gt; will be treated as JSON, just like they already are in the {{ns:2}} and {{ns:8}} namespaces. [https://phabricator.wikimedia.org/T144475]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:30, 27 September 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23860085 --&gt;

== Tech News: 2022-40 ==

&lt;section begin=&quot;technews-2022-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/40|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps can now show geopoints from Wikidata, via QID or SPARQL query. Previously, this was only possible for geoshapes and geolines. [https://phabricator.wikimedia.org/T307695] [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Geopoints_via_QID]
* The [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award 2022]] is looking for nominations. You can recommend tools until 12 October.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-05|en}}. It will be on all wikis from {{#time:j xg|2022-10-06|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|Talk pages on the mobile site]] will change at the Arabic, Bangla, Chinese, French, Haitian Creole, Hebrew, Korean, and Vietnamese Wikipedias.  They should be easier to use and provide more information. (Last week's release was delayed) [https://phabricator.wikimedia.org/T318302] [https://www.mediawiki.org/wiki/Talk_pages_project/Mobile]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;scribunto-console&lt;/code&gt;&lt;/bdi&gt; API module will require a [[mw:Special:MyLanguage/API:Tokens|CSRF token]]. This module is documented as internal and use of it is not supported. [[phab:T212071|[5]]] 
* The Vector 2022 skin will become the default across the smallest Wikimedia projects. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#Deployment_plan_and_timeline|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:23, 4 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23885489 --&gt;

== Tech News: 2022-41 ==

&lt;section begin=&quot;technews-2022-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/41|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-12|en}}. It will be on all wikis from {{#time:j xg|2022-10-13|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* On some wikis, [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps in full size view will be able to display nearby articles. After a feedback period, more wikis will follow. [https://phabricator.wikimedia.org/T316782][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Nearby_articles]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W41&quot;/&gt;

14:08, 10 October 2022 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23912412 --&gt;

== Tech News: 2022-42 ==

&lt;section begin=&quot;technews-2022-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/42|Translations]] are available.

'''Recent changes'''
* The recently implemented feature of [[phab:T306883|article thumbnails in Special:Search]] will be limited to Wikipedia projects only. Further details are in [[phab:T320510|T320510]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Structured_Data_Across_Wikimedia/Search_Improvements]
* A bug that caused problems in loading article thumbnails in Special:Search has been fixed. Further details are in [[phab:T320406|T320406]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-19|en}}. It will be on all wikis from {{#time:j xg|2022-10-20|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Lua module authors can use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Extension:Scribunto/Lua_reference_manual#mw.loadJsonData|mw.loadJsonData()]]&lt;/code&gt;&lt;/bdi&gt; to load data from JSON pages. [https://phabricator.wikimedia.org/T217500]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Lua module authors can enable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Extension:Scribunto/Lua_reference_manual#Strict_library|require( &quot;strict&quot; )]]&lt;/code&gt;&lt;/bdi&gt; to add errors for some possible code problems. This replaces &quot;[[wikidata:Q16748603|Module:No globals]]&quot; on most wikis. [https://phabricator.wikimedia.org/T209310]

'''Future changes'''
* The [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated at most wikis. The &quot;{{int:discussiontools-replylink}}&quot; button will look different after this change. [https://phabricator.wikimedia.org/T320683]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:46, 17 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23943992 --&gt;

== Tech News: 2022-43 ==

&lt;section begin=&quot;technews-2022-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/43|Translations]] are available.

'''Recent changes'''
* There have been some minor visual fixes in Special:Search, regarding audio player alignment and image placeholder height. Further details are in [[phab:T319230|T319230]].
* On Wikipedias, a new [[Special:Preferences#mw-prefsection-searchoptions|preference]] has been added to hide article thumbnails in Special:Search. Full details are in [[phab:T320337|T320337]].

'''Problems'''
* Last week, three wikis ({{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ruwiki/en}}) had read-only access for 25 minutes. This was caused by a hardware problem. [https://phabricator.wikimedia.org/T320990]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-26|en}}. It will be on all wikis from {{#time:j xg|2022-10-27|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-10-25|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2022-10-27|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-aswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-banwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-barwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bat smgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bclwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-be x oldwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-biwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bjnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bpywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-brwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bugwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bxrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304549] 
* Starting on Wednesday October 26, 2022, the list of mentors will be upgraded [[d:Q14339834 | at wikis where Growth mentorship is available]]. The mentorship system will continue to work as it does now. The signup process [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list#add|will be replaced]], and a new management option will be provided. Also, this change simplifies [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list#create|the creation of mentorship systems at Wikipedias]]. [https://phabricator.wikimedia.org/T314858][https://phabricator.wikimedia.org/T310905][https://www.mediawiki.org/wiki/Special:MyLanguage/Growth/Structured_mentor_list]
* Pages with titles that start with a lower-case letter according to Unicode 11 will be renamed or deleted. There is a list of affected pages at &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:Unicode 11 case map migration]]&lt;/bdi&gt;. More information can be found at [[phab:T292552|T292552]].
* The Vector 2022 skin will become the default across the smallest Wikipedias. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#smallest-1|Learn more]].

'''Future changes'''
* The [[mw:Special:MyLanguage/Talk pages project/Replying|Reply tool]] and [[mw:Special:MyLanguage/Talk pages project/New discussion|New Topic tool]] will soon get a [[mw:Special:MyLanguage/VisualEditor/Special characters|special characters menu]]. [https://phabricator.wikimedia.org/T249072]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:22, 24 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23975411 --&gt;

== Tech News: 2022-44 ==

&lt;section begin=&quot;technews-2022-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/44|Translations]] are available.

'''Recent changes'''
* When using keyboard navigation on a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map, the focus will become more visible. [https://phabricator.wikimedia.org/T315997]
* In {{#special:RecentChanges}}, you can now hide the log entries for new user creations with the filter for &quot;{{int:rcfilters-filter-newuserlogactions-label}}&quot;. [https://phabricator.wikimedia.org/T321155]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-02|en}}. It will be on all wikis from {{#time:j xg|2022-11-03|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* The [[mw:Special:MyLanguage/Help:Extension:Kartographer|maps dialog]] in VisualEditor now has some help texts. [https://phabricator.wikimedia.org/T318818]
* It is now possible to select the language of a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor via a dropdown menu. [https://phabricator.wikimedia.org/T318817]
* It is now possible to add a caption to a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor. [https://phabricator.wikimedia.org/T318815]
* It is now possible to hide the frame of a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor. [https://phabricator.wikimedia.org/T318813]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:15, 31 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23977539 --&gt;

== Tech News: 2022-45 ==

&lt;section begin=&quot;technews-2022-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/45|Translations]] are available.

'''Recent changes'''
* An updated version of the [[m:Special:MyLanguage/EventCenter/Registration|Event Registration]] tool is now available for testing at [[testwiki:|testwiki]] and [[test2wiki:| test2wiki]]. The tool provides features for event organizers and participants. Your feedback is welcome at our [[m:Talk:Campaigns/Foundation Product Team/Registration|project talkpage]]. More information about [[m:Campaigns/Foundation Product Team/Registration|the project]] is available. [https://phabricator.wikimedia.org/T318592]

'''Problems'''
* Twice last week, for about 45 minutes, some files and thumbnails failed to load and uploads failed, mostly for logged-in users. The cause is being investigated and an incident report will be available soon.

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:32, 8 November 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24001035 --&gt;

== Tech News: 2022-46 ==

&lt;section begin=&quot;technews-2022-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/46|Translations]] are available.

'''Recent changes'''
* At Wikidata, an interwiki link can now point to a redirect page if certain conditions are met. This new feature is called [[wikidata:Special:MyLanguage/Wikidata:Sitelinks_to_redirects|sitelinks to redirects]]. It is needed when one wiki uses one page to cover multiple concepts but another wiki uses more pages to cover the same concepts. Your [[wikidata:Special:MyLanguage/Wikidata talk:Sitelinks to redirects|feedback on the talkpage]] of the new proposed guideline is welcome. [https://phabricator.wikimedia.org/T278962]
* The &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikinews.org/ www.wikinews.org]&lt;/span&gt;, &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikiversity.org/ www.wikiversity.org]&lt;/span&gt;, and &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikivoyage.org/ www.wikivoyage.org]&lt;/span&gt; portal pages now use an automated update system. [https://phabricator.wikimedia.org/T273179]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-16|en}}. It will be on all wikis from {{#time:j xg|2022-11-17|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* There will be a new link to directly &quot;Edit template data&quot; on Template pages. [https://phabricator.wikimedia.org/T316759]

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Wikis where mobile [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] are enabled ([[mw:Special:MyLanguage/Talk pages project/Deployment Status|these ones]]) will soon use full CSS styling to display any templates that are placed at the top of talk pages. To adapt these “talk page boxes” for narrow mobile devices you can use media queries, such as in [https://en.wikipedia.org/w/index.php?title=Module:Message_box/tmbox.css&amp;oldid=1097618699#L-69 this example]. [https://phabricator.wikimedia.org/T312309]
* Starting in January 2023, [[m:Special:MyLanguage/Community Tech|Community Tech]] will be [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2023 Changes Update|running the Community Wishlist Survey (CWS) every two years]]. This means that in 2024, there will be no new proposals or voting.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:54, 14 November 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24071290 --&gt;

== Tech News: 2022-47 ==

&lt;section begin=&quot;technews-2022-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/47|Translations]] are available.

'''Recent changes'''
* The display of non-free media in the search bar and for article thumbnails in Special:Search has been deactivated. Further details are in [[phab:T320661|T320661]].

'''Changes later this week'''
* There is no new MediaWiki version this week.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-11-22|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]) and on {{#time:j xg|2022-11-24|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:22, 21 November 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24071290 --&gt;

== Tech News: 2022-48 ==

&lt;section begin=&quot;technews-2022-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/48|Translations]] are available.

'''Recent changes'''
* A new preference, “Enable limited width mode”, has been added to the [[Special:Preferences#mw-prefsection-rendering|Vector 2022 skin]]. The preference is also available as a toggle on every page if your monitor is 1600 pixels or wider. It allows for increasing the width of the page for logged-out and logged-in users. [https://phabricator.wikimedia.org/T319449]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-30|en}}. It will be on all wikis from {{#time:j xg|2022-12-01|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-11-29|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-12-01|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]).
* Mathematical formulas shown in SVG image format will no longer have PNG fall-backs for browsers that don't support them. This is part of work to modernise the generation system. Showing only PNG versions was the default option until in February 2018. [https://lists.wikimedia.org/hyperkitty/list/wikimedia-l@lists.wikimedia.org/message/3BGOKWJIZGL4TC4HJ22ICRU2SEPWGCR4/][https://phabricator.wikimedia.org/T311620][https://phabricator.wikimedia.org/T186327]
* On [[phab:P40224|some wikis]] that use flagged revisions, [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs#Special:Contributions|a new checkbox will be added]] to Special:Contributions that enables you to see only the [[mw:Special:MyLanguage/Help:Pending changes|pending changes]] by a user. [https://phabricator.wikimedia.org/T321445]

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] How media is structured in the parser's HTML output will change early next week at [https://wikitech.wikimedia.org/wiki/Deployments/Train#Wednesday group1 wikis] (but not Wikimedia Commons or Meta-Wiki). This change improves the accessibility of content, and makes it easier to write related CSS. You may need to update your site-CSS, or userscripts and gadgets. There are [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Media_structure/FAQ|details on what code to check, how to update the code, and where to report any related problems]]. [https://phabricator.wikimedia.org/T314318]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:03, 28 November 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24114342 --&gt;

== Tech News: 2022-49 ==

&lt;section begin=&quot;technews-2022-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/49|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Wikisources use a tool called ProofreadPage. ProofreadPage uses OpenSeadragon which is an open source tool. The OpenSeadragon JavaScript API has been significantly re-written to support dynamically loading images. The functionality provided by the older version of the API should still work but it is no longer supported. User scripts and gadgets should migrate over to the newer version of the API. The functionality provided by the newer version of the API is [[mw:Extension:Proofread_Page/Page_viewer#JS_API|documented on MediaWiki]]. [https://phabricator.wikimedia.org/T308098][https://www.mediawiki.org/wiki/Extension:Proofread_Page/Edit-in-Sequence]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-12-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-12-07|en}}. It will be on all wikis from {{#time:j xg|2022-12-08|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:41, 6 December 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24151590 --&gt;

== Tech News: 2022-50 ==

&lt;section begin=&quot;technews-2022-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/50|Translations]] are available.

'''Recent changes'''
* An [[mw:Special:MyLanguage/Talk pages project/Mobile|A/B test has begun]] at 15 Wikipedias for [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|DiscussionTools on mobile]].  Half of the editors on the [[mw:Reading/Web/Mobile|mobile web site]] will have access to the {{int:discussiontools-replybutton}} tool and other features. [https://phabricator.wikimedia.org/T321961]
* The character &lt;code&gt;=&lt;/code&gt; cannot be used in new usernames, to make usernames work better with templates. Existing usernames are not affected. [https://phabricator.wikimedia.org/T254045]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-12-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-12-14|en}}. It will be on all wikis from {{#time:j xg|2022-12-15|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The HTML markup used by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] to [[mw:Special:MyLanguage/Talk_pages_project/Usability#Phase_1:_Topic_containers|show discussion metadata below section headings]] will be inserted after these headings, not inside of them. This change improves the accessibility of discussion pages for screen reader software. [https://phabricator.wikimedia.org/T314714]

'''Events'''
* The fourth edition of the [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award]] will happen online on [https://zonestamp.toolforge.org/1671210002 Friday 16 December 2022 at 17:00 UTC]! The event will be live-streamed on YouTube in the [https://www.youtube.com/user/watchmediawiki MediaWiki channel] and added to Commons afterwards.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:34, 12 December 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24216570 --&gt;

== Tech News: 2022-51 ==

&lt;section begin=&quot;technews-2022-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/51|Translations]] are available.

'''Tech News'''
* Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 9 January 2023.

'''Recent changes'''
* On a user's contributions page, you can filter it for edits with a tag like 'reverted'. Now, you can also filter for all edits that are not tagged like that. This was part of a Community Wishlist 2022 request. [https://phabricator.wikimedia.org/T119072]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new function has been used for gadget developers to add content underneath the title on article pages. This is considered a stable API that should work across all skins. [[mw:Special:MyLanguage/ResourceLoader/Core_modules#addSubtitle|Documentation is available]]. [https://phabricator.wikimedia.org/T316830]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[test2wiki:|One of our test wikis]] is now being served from a new infrastructure powered by [[w:Kubernetes|Kubernetes]] ([[wikitech:MediaWiki On Kubernetes|read more]]). More Wikis will switch to this new infrastructure in early 2023. Please test and let us know of any issues. [https://phabricator.wikimedia.org/T290536]

'''Problems'''
* Last week, all wikis had no edit access for 9 minutes. This was caused by a database problem. [https://wikitech.wikimedia.org/wiki/Incidents/2022-12-13_sessionstore]

'''Changes later this week'''
* There is no new MediaWiki version this week or next week.
* The word &quot;{{int:discussiontools-replybutton}}&quot; is very short in some languages, such as Arabic (&quot;&lt;bdi lang=&quot;ar&quot;&gt;ردّ&lt;/bdi&gt;&quot;). This makes the {{int:discussiontools-preference-label}} button on talk pages difficult to use. An arrow icon will be added to those languages. This will only be visible to editors who have the [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] turned on. [https://www.mediawiki.org/wiki/Talk_pages_project/Usability#Status] [https://phabricator.wikimedia.org/T323537]

'''Future changes'''
* Edits can be automatically &quot;tagged&quot; by the system software or the {{int:Abusefilter}} system. Those tags link to a help page about the tags. Soon they will also link to Recent Changes to let you see other edits tagged this way. This was a Community Wishlist 2022 request. [https://phabricator.wikimedia.org/T301063]
* The Trust &amp; Safety tools team [[m:Special:MyLanguage/Private Incident Reporting System/Timeline and Updates|have shared new plans]] for building the Private Incident Reporting System. The system will make it easier for editors to ask for help if they are harassed or abused. 
* [[m:Special:MyLanguage/Community Wishlist Survey 2021/Real Time Preview for Wikitext|Realtime Preview for Wikitext]] is coming out of beta as an enabled feature for every user of the 2010 Wikitext [[mw:Special:MyLanguage/Editor|editor]] in the week of January 9, 2023. It will be available to use via the toolbar in the 2010 Wikitext editor. The feature was the 4th most popular wish of the Community Wishlist Survey 2021.

'''Events'''
* You can now [[mw:Special:MyLanguage/Wikimedia Hackathon 2023/Participate|register for the Wikimedia Hackathon 2023]], taking place on May 19–21 in Athens, Greece. You can also apply for a scholarship until January 14th.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:00, 20 December 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24258101 --&gt;

== Tech News: 2023-02 ==

&lt;section begin=&quot;technews-2023-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/02|Translations]] are available.

'''Recent changes'''
* You can use tags to filter edits in the recent changes feed or on your watchlist. You can now use tags to filter out edits you don't want to see. Previously you could only use tags to focus on the edits with those tags. [https://phabricator.wikimedia.org/T174349]
* [[Special:WhatLinksHere|Special:WhatLinksHere]] shows all pages that link to a specific page. There is now a [https://wlh.toolforge.org prototype] for how to sort those pages alphabetically. You can see the discussion in the [[phab:T4306|Phabricator ticket]].
* You can now use the [[mw:Special:MyLanguage/Extension:Thanks|thanks]] function on your watchlist and the user contribution page. [https://phabricator.wikimedia.org/T51541]
* A wiki page can be moved to give it a new name. You can now get a dropdown menu with common reasons when you move a page. This is so you don't have to write the explanation every time. [https://phabricator.wikimedia.org/T325257]
* [[m:Special:MyLanguage/Matrix.org|Matrix]] is a chat tool. You can now use &lt;code&gt;matrix:&lt;/code&gt; to create Matrix links on wiki pages. [https://phabricator.wikimedia.org/T326021]
* You can filter out translations when you look at the recent changes on multilingual wikis. This didn't hide translation pages. You can now also hide subpages which are translation pages. [https://phabricator.wikimedia.org/T233493]

'''Changes later this week'''
* [[m:Special:MyLanguage/Real Time Preview for Wikitext|Realtime preview for wikitext]] is a tool which lets editors preview the page when they edit wikitext. It will be enabled for all users of the 2010 wikitext editor. You will find it in the editor toolbar.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2023-01-10|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2023-01-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-11|en}}. It will be on all wikis from {{#time:j xg|2023-01-12|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W02&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:07, 10 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24342971 --&gt;

== Tech News: 2023-03 ==

&lt;section begin=&quot;technews-2023-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/03|Translations]] are available.

'''Problems'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The URLs in &quot;{{int:last}}&quot; links on page history now contain &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;diff=prev&amp;oldid=[revision ID]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; in place of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;diff=[revision ID]&amp;oldid=[revision ID]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This is to fix a problem with links pointing to incorrect diffs when history was filtered by a tag. Some user scripts may break as a result of this change. [https://phabricator.wikimedia.org/T243569]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-18|en}}. It will be on all wikis from {{#time:j xg|2023-01-19|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Some [[mw:Special:MyLanguage/Talk pages project/Usability|changes to the appearance of talk pages]] have only been available on &lt;code&gt;{{ns:1}}:&lt;/code&gt; and &lt;code&gt;{{ns:3}}:&lt;/code&gt; namespaces.  These will be extended to other talk namespaces, such as &lt;code&gt;{{ns:5}}:&lt;/code&gt;.  They will continue to be unavailable in non-talk namespaces, including &lt;code&gt;{{ns:4}}:&lt;/code&gt; pages (e.g., at the Village Pump).  You can [[Special:Preferences#mw-prefsection-editing-discussion|change your preferences]] ([[Special:Preferences#mw-prefsection-betafeatures|beta feature]]). [https://phabricator.wikimedia.org/T325417]
*On Wikisources, when an image is zoomed or panned in the Page: namespace, the same zoom and pan settings will be remembered for all Page: namespace pages that are linked to a particular Index: namespace page. [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/ProofreadPage/+/868841]
* The Vector 2022 skin will become the default for the English Wikipedia desktop users. The change will take place on January 18 at 15:00 UTC. [[:en:w:Wikipedia:Vector 2022|Learn more]].

'''Future changes'''
* The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]], which invites contributors to make technical proposals and vote for tools and improvements, starts next week on 23 January 2023 at 18:00 UTC. You can start drafting your proposals in [[m:Community Wishlist Survey/Sandbox|the CWS sandbox]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:10, 17 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24381020 --&gt;

== Tech News: 2023-04 ==

&lt;section begin=&quot;technews-2023-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/04|Translations]] are available.

'''Problems'''
* Last week, for ~15 minutes, all wikis were unreachable for logged-in users and non-cached pages. This was caused by a timing issue. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-17_MediaWiki]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-25|en}}. It will be on all wikis from {{#time:j xg|2023-01-26|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* If you have the Beta Feature for [[mw:Special:MyLanguage/Talk pages project|DiscussionTools]] enabled, the appearance of talk pages will add more information about discussion activity. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/Usability#Status][https://phabricator.wikimedia.org/T317907]
* The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]] (CWS), which invites contributors to make technical proposals and vote for tools and improvements, starts on Monday 23 January 2023 at [https://zonestamp.toolforge.org/1674496814 18:00 UTC]. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:46, 23 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24418874 --&gt;

== Tech News: 2023-05 ==

&lt;section begin=&quot;technews-2023-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/05|Translations]] are available.

'''Problems'''
* Last week, for ~15 minutes, some users were unable to log in or edit pages. This was caused by a problem with session storage. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-24_sessionstore_quorum_issues]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-01|en}}. It will be on all wikis from {{#time:j xg|2023-02-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Wikis that use localized numbering schemes for references need to add new CSS. This will help to show citation numbers the same way in all reading and editing modes. If your wiki would prefer to do it yourselves, please see the [[mw:Special:MyLanguage/Parsoid/Parser Unification/Cite CSS|details and example CSS to copy from]], and also add your wiki to the list. Otherwise, the developers will directly help out starting the week of February 5.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:05, 31 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24455949 --&gt;

== Tech News: 2023-06 ==

&lt;section begin=&quot;technews-2023-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/06|Translations]] are available.

'''Recent changes'''
* In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], logged-out users using the full-width toggle will be able to see the setting of their choice even after refreshing pages or opening new ones. This only applies to wikis where Vector 2022 is the default. [https://phabricator.wikimedia.org/T321498]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-08|en}}. It will be on all wikis from {{#time:j xg|2023-02-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Previously, we announced when some wikis would be in read-only for a few minutes because of a switch of their main database. These switches will not be announced any more, as the read-only time has become non-significant. Switches will continue to happen at 7AM UTC on Tuesdays and Thursdays. [https://phabricator.wikimedia.org/T292543#8568433]
* Across all the wikis, in the Vector 2022 skin, logged-in users will see the page-related links such as &quot;What links here&quot; in a [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Features/Page_tools|new side menu]]. It will be displayed on the other side of the screen. This change had previously been made on Czech, English, and Vietnamese Wikipedias. [https://phabricator.wikimedia.org/T328692]
*[[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] will stop receiving new proposals on [https://zonestamp.toolforge.org/1675706431 Monday, 6 February 2023, at 18:00 UTC]. Proposers should complete any edits by then, to give time for [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translations]] and review. Voting will begin on Friday, 10 February.

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadgets and user scripts will be changing to load on desktop and mobile sites. Previously they would only load on the desktop site. It is recommended that wiki administrators audit the [[MediaWiki:Gadgets-definition|gadget definitions]] prior to this change, and add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;skins=…&lt;/code&gt;&lt;/bdi&gt; for any gadgets which should not load on mobile. [https://phabricator.wikimedia.org/T328610 More details are available].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 10:21, 6 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24491749 --&gt;

== Tech News: 2023-07 ==

&lt;section begin=&quot;technews-2023-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/07|Translations]] are available.

'''Problems'''
* On wikis where patrolled edits are enabled, changes made to the [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|mentor list]] by autopatrolled mentors are not correctly marked as patrolled. It will be fixed later this week. [https://phabricator.wikimedia.org/T328444]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-15|en}}. It will be on all wikis from {{#time:j xg|2023-02-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* The Reply tool and other parts of [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|DiscussionTools]] will be deployed for all editors using the mobile site.  You can [[mw:Special:MyLanguage/Talk_pages_project/Mobile#Status_Updates|read more about this decision]]. [https://phabricator.wikimedia.org/T298060]

'''Future changes'''
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:48, 14 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24540832 --&gt;

== Tech News: 2023-08 ==

&lt;section begin=&quot;technews-2023-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/08|Translations]] are available.

'''Problems'''
* Last week, during planned maintenance of Cloud Services, unforeseen complications forced the team to turn off all tools for 2–3 hours to prevent data corruption. Work is ongoing to prevent similar problems in the future. [https://phabricator.wikimedia.org/T329535] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-22|en}}. It will be on all wikis from {{#time:j xg|2023-02-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
*The voting phase for the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] ends on [https://zonestamp.toolforge.org/1677261621 24 February at 18:00 UTC]. The results of the survey will be announced on 28 February.

'''Future changes'''
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:57, 21 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24570514 --&gt;

== Tech News: 2023-09 ==

&lt;section begin=&quot;technews-2023-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/09|Translations]] are available.

'''Problems'''
* Last week, in some areas of the world, there were problems with loading pages for 20 minutes and saving edits for 55 minutes. These issues were caused by a problem with our caching servers due to unforseen events during a routine maintenance task. [https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_wiki_outage][https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_read_only] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-01|en}}. It will be on all wikis from {{#time:j xg|2023-03-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:47, 27 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24634242 --&gt;

== Tech News: 2023-10 ==

&lt;section begin=&quot;technews-2023-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/10|Translations]] are available.

'''Recent changes'''
* The Community Wishlist Survey 2023 edition has been concluded. Community Tech has [[m:Special:MyLanguage/Community Wishlist Survey 2023/Results|published the results]] of the survey and will provide an update on what is next in April 2023.
* On wikis which use [[mw:Special:MyLanguage/Writing_systems|LanguageConverter]] to handle multiple writing systems, articles which used custom conversion rules in the wikitext (primarily on Chinese Wikipedia) would have these rules applied inconsistently in the table of contents, especially in the Vector 2022 skin. This has now been fixed. [https://phabricator.wikimedia.org/T306862]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-08|en}}. It will be on all wikis from {{#time:j xg|2023-03-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* A search system has been added to the [[Special:Preferences|Preferences screen]]. This will let you find different options more easily. Making it work on mobile devices will happen soon. [https://phabricator.wikimedia.org/T313804]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:49, 6 March 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24676916 --&gt;

== Tech News: 2023-11 ==

&lt;section begin=&quot;technews-2023-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/11|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-15|en}}. It will be on all wikis from {{#time:j xg|2023-03-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-cbk_zamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cdowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cebwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ckbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-csbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542][https://phabricator.wikimedia.org/T304550] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:20, 13 March 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24700189 --&gt;

== Tech News: 2023-12 ==

&lt;section begin=&quot;technews-2023-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/12|Translations]] are available.

'''Problems'''
* Last week, some users experienced issues loading image thumbnails. This was due to incorrectly cached images. [https://phabricator.wikimedia.org/T331820]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-22|en}}. It will be on all wikis from {{#time:j xg|2023-03-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A link to the user's [[{{#special:CentralAuth}}]] page will appear on [[{{#special:Contributions}}]] — some user scripts which previously added this link may cause conflicts. This feature request was [[:m:Community Wishlist Survey 2023/Admins and patrollers/Add link to CentralAuth on Special:Contributions|voted #17 in the 2023 Community Wishlist Survey]].
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[{{#special:AbuseFilter}}]] edit window will be resizable and larger by default. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Make the AbuseFilter edit window resizable and larger by default|voted #80 in the 2023 Community Wishlist Survey]].
* There will be a new option for Administrators when they are unblocking a user, to add the unblocked user’s user page to their watchlist. This will work both via [[{{#special:Unblock}}]] and via the API. [https://phabricator.wikimedia.org/T257662]

'''Meetings'''
* You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1679677204 24 March at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia Apps/Office Hours|details and how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:25, 21 March 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24732558 --&gt;

== Tech News: 2023-13 ==

&lt;section begin=&quot;technews-2023-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/13|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] condition limit was increased from 1000 to 2000. [https://phabricator.wikimedia.org/T309609]
* [[:m:Special:MyLanguage/Global AbuseFilter#Locally disabled actions|Some Global AbuseFilter]] actions will no longer apply to local projects. [https://phabricator.wikimedia.org/T332521]
* Desktop users are now able to subscribe to talk pages by clicking on the {{int:discussiontools-newtopicssubscription-button-subscribe-label}} link in the {{int:toolbox}} menu. If you subscribe to a talk page, you receive [[mw:Special:MyLanguage/Notifications|notifications]] when new topics are started on that talk page. This is separate from putting the page on your watchlist or subscribing to a single discussion. [https://phabricator.wikimedia.org/T263821]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-29|en}}. It will be on all wikis from {{#time:j xg|2023-03-30|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''Future changes'''
* You will be able to choose [[mw:Special:MyLanguage/VisualEditor/Diffs|visual diffs]] on all [[m:Special:MyLanguage/Help:Page history|history pages]] at the Wiktionaries and Wikipedias. [https://phabricator.wikimedia.org/T314588]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The legacy [[mw:Mobile Content Service|Mobile Content Service]] is going away in July 2023. Developers are encouraged to switch to Parsoid or another API before then to ensure service continuity. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/4MVQQTONJT7FJAXNVOFV3WWVVMCHRINE/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:13, 28 March 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24780854 --&gt;

== Tech News: 2023-14 ==

&lt;section begin=&quot;technews-2023-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/14|Translations]] are available.

'''Recent changes'''
* The system for automatically creating categories for the [[mw:Special:MyLanguage/Extension:Babel|Babel]] extension has had several important changes and fixes. One of them allows you to insert templates for automatic category descriptions on creation, allowing you to categorize the new categories. [https://phabricator.wikimedia.org/T211665][https://phabricator.wikimedia.org/T64714][https://phabricator.wikimedia.org/T170654][https://phabricator.wikimedia.org/T184941][https://phabricator.wikimedia.org/T33074]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-05|en}}. It will be on all wikis from {{#time:j xg|2023-04-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Some older [[w:en:Web browser|Web browsers]] will stop being able to use [[w:en:JavaScript|JavaScript]] on Wikimedia wikis from this week. This mainly affects users of Internet Explorer 11. If you have an old web browser on your computer you can try to upgrade to a newer version. [https://phabricator.wikimedia.org/T178356]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The deprecated &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.hoverIntent&lt;/code&gt;&lt;/bdi&gt; module has been removed. This module could be used by gadgets and user scripts, to create an artificial delay in how JavaScript responds to a hover event. Gadgets and user scripts should now use jQuery &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;hover()&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;on()&lt;/code&gt;&lt;/bdi&gt; instead. Examples can be found in the [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#jquery.hoverIntent|migration guide]]. [https://phabricator.wikimedia.org/T311194]
* Some of the links in [[{{#special:SpecialPages}}]] will be re-arranged. There will be a clearer separation between links that relate to all users, and links related to your own user account. [https://phabricator.wikimedia.org/T333242]
* You will be able to hide the [[mw:Special:MyLanguage/Talk pages project/Replying|Reply button]] in archived discussion pages with a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__ARCHIVEDTALK__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; magic word.  There will also be a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.mw-archivedtalk&lt;/code&gt;&lt;/bdi&gt; CSS class for hiding the Reply button in individual sections on a page. [https://phabricator.wikimedia.org/T249293][https://phabricator.wikimedia.org/T295553][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/DiscussionTools/+/738221]

'''Future changes'''
* The Vega software that creates data visualizations in pages, such as graphs, will be upgraded to the newest version in the future. Graphs that still use the very old version 1.5 syntax may stop working properly. Most existing uses have been found and updated, but you can help to check, and to update any local documentation. [[phab:T260542|Examples of how to find and fix these graphs are available]]. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:39, 3 April 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24820268 --&gt;

== Tech News: 2023-15 ==

&lt;section begin=&quot;technews-2023-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/15|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] In the visual editor, it is now possible to edit captions of images in galleries without opening the gallery dialog. This feature request was [[:m:Community Wishlist Survey 2023/Editing/Editable gallery captions in Visual Editor|voted #61 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T190224]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] You can now receive notifications when another user edits your user page. See the &quot;{{int:Echo-category-title-edit-user-page}}&quot; option in [[Special:Preferences#mw-prefsection-echo|your Preferences]]. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Notifications for user page edits|voted #3 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T3876]

'''Problems'''
* There was a problem with all types of CentralNotice banners still being shown to logged-in users even if they had [[Special:Preferences#mw-prefsection-centralnotice-banners|turned off]] specific banner types. This has now been fixed. [https://phabricator.wikimedia.org/T331671]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-12|en}}. It will be on all wikis from {{#time:j xg|2023-04-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-arywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dinwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-elwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-emlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-etwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-euwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-extwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tumwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ffwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiu_vrowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-furwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gcrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gomwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gotwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-guwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304551][https://phabricator.wikimedia.org/T308133]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:05, 10 April 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24851886 --&gt;

== Tech News: 2023-16 ==

&lt;section begin=&quot;technews-2023-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/16|Translations]] are available.

'''Recent changes'''
* You can now see [[mw:Special:MyLanguage/Help:Extension:Kartographer#Show_nearby_articles|nearby articles on a Kartographer map]] with the button for the new feature &quot;{{int:Kartographer-sidebar-nearbybutton}}&quot;. Six wikis have been testing this feature since October. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Nearby_articles#Implementation][https://phabricator.wikimedia.org/T334079]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] page now has links for &quot;{{int:globalwatchlist-markpageseen}}&quot; for each entry. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Button to mark a single change as read in the global watch list|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334246]

'''Problems'''
* At Wikimedia Commons, some thumbnails have not been getting replaced correctly after a new version of the image is uploaded. This should be fixed later this week. [https://phabricator.wikimedia.org/T331138][https://phabricator.wikimedia.org/T333042]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] For the last few weeks, some external tools had inconsistent problems with logging-in with OAuth. This has now been fixed. [https://phabricator.wikimedia.org/T332650]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-19|en}}. It will be on all wikis from {{#time:j xg|2023-04-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:54, 18 April 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24881071 --&gt;

== Tech News: 2023-17 ==

&lt;section begin=&quot;technews-2023-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/17|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The date-selection menu on pages such as [[{{#special:Contributions}}]] will now show year-ranges that are in the current and past decade, instead of the current and future decade. This feature request was [[m:Community Wishlist Survey 2023/Miscellaneous/Change year range shown in date selection popup|voted #145 in the 2023 Community Wishlist Survey]].  [https://phabricator.wikimedia.org/T334316]

'''Problems'''
* Due to security issues with the [[mw:Special:MyLanguage/Extension:Graph|Graph extension]], graphs have been disabled in all Wikimedia projects. Wikimedia Foundation teams are working to respond to these vulnerabilities. [https://phabricator.wikimedia.org/T334940]
* For a few days, it was not possible to save some kinds of edits on the mobile version of a wiki. This has been fixed. [https://phabricator.wikimedia.org/T334797][https://phabricator.wikimedia.org/T334799][https://phabricator.wikimedia.org/T334794]

'''Changes later this week'''
* All wikis will be read-only for a few minutes on April 26. This is planned for [https://zonestamp.toolforge.org/1682517653 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-26|en}}. It will be on all wikis from {{#time:j xg|2023-04-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* The Editing team plans an A/B test for [[mw:Special:MyLanguage/Talk pages project/Usability|a usability analysis of the Talk page project]]. The [[mw:Special:MyLanguage/Talk pages project/Usability/Analysis|planned measurements are available]]. Your wiki [[phab:T332946|may be invited to participate]]. Please suggest improvements to the measurement plan at [[mw:Talk:Talk pages project/Usability|the discussion page]].
* [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2023-2024|The Wikimedia Foundation annual plan 2023-2024 draft is open for comment and input]] until May 19. The final plan will be published in July 2023 on Meta-wiki.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:03, 24 April 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24933592 --&gt;

== Tech News: 2023-18 ==

&lt;section begin=&quot;technews-2023-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/18|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the French and Italian Wikipedias. More languages will be added in the near future. This is part of the [[m:Community Wishlist Survey 2023/Reading/Extend &quot;Who Wrote That?&quot; tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T243711][https://phabricator.wikimedia.org/T270490][https://phabricator.wikimedia.org/T334891]
* The [[:commons:Special:MyLanguage/Commons:Video2commons|Video2commons]] tool has been updated. This fixed several bugs related to YouTube uploads. [https://github.com/toolforge/video2commons/pull/162/commits]
* The [[{{#special:Preferences}}]] page has been redesigned on mobile web. The new design makes it easier to browse the different categories and settings at low screen widths. You can also now access the page via a link in the Settings menu in the mobile web sidebar. [https://www.mediawiki.org/wiki/Moderator_Tools/Content_moderation_on_mobile_web/Preferences]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-03|en}}. It will be on all wikis from {{#time:j xg|2023-05-04|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:45, 2 May 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24966974 --&gt;

== Tech News: 2023-19 ==

&lt;section begin=&quot;technews-2023-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/19|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] Last week, Community Tech released the first update for providing [[m:Special:MyLanguage/Community Wishlist Survey 2022/Better diff handling of paragraph splits|better diffs]], the #1 request in the 2022 Community Wishlist Survey. [[phab:T324759|This update]] adds legends and tooltips to inline diffs so that users unfamiliar with the blue and yellow highlights can better understand the type of edits made.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] When you close an image that is displayed via MediaViewer, it will now return to the wiki page instead of going back in your browser history. This feature request was [[m:Community Wishlist Survey 2023/Reading/Return to the article when closing the MediaViewer|voted #65 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T236591]
* The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|SyntaxHighlight]] extension now supports &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wikitext&lt;/code&gt;&lt;/bdi&gt; as a selected language. Old alternatives that were used to highlight wikitext, such as &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;html5&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;moin&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;html+handlebars&lt;/code&gt;&lt;/bdi&gt;, can now be replaced. [https://phabricator.wikimedia.org/T29828]
* [[mw:Special:MyLanguage/Manual:Creating pages with preloaded text|Preloading text to new pages/sections]] now supports preloading from localized MediaWiki interface messages. [https://cs.wikipedia.org/wiki/User_talk:Martin_Urbanec_(WMF)?action=edit&amp;section=new&amp;preload=MediaWiki:July Here is an example] at the {{int:project-localized-name-cswiki/en}} that uses &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;preload=MediaWiki:July&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T330337]

'''Problems'''
* Graph Extension update: Foundation developers have completed upgrading the visualization software to Vega5. Existing community graphs based on Vega2 are no longer compatible. Communities need to update local graphs and templates, and shared lua modules like &lt;bdi lang=&quot;de&quot; dir=&quot;ltr&quot;&gt;[[:de:Modul:Graph]]&lt;/bdi&gt;. The [https://vega.github.io/vega/docs/porting-guide/ Vega Porting guide] provides the most comprehensive detail on migration from Vega2 and [https://www.mediawiki.org/w/index.php?title=Template:Graph:PageViews&amp;action=history here is an example migration]. Vega5 has currently just been enabled on mediawiki.org to provide a test environment for communities. [https://phabricator.wikimedia.org/T334940#8813922]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-10|en}}. It will be on all wikis from {{#time:j xg|2023-05-11|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Until now, all new OAuth apps went through manual review. Starting this week, apps using identification-only or basic authorizations will not require review. [https://phabricator.wikimedia.org/T67750]

'''Future changes'''
* During the next year, MediaWiki will stop using IP addresses to identify logged-out users, and will start automatically assigning unique temporary usernames. Read more at [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates|IP Editing: Privacy Enhancement and Abuse Mitigation/Updates]]. You can [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation#What should it look like?|join the discussion]] about the [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates#What will temporary usernames look like?|format of the temporary usernames]]. [https://phabricator.wikimedia.org/T332805]
* There will be an [[:w:en:A/B testing|A/B test]] on 10 Wikipedias where the Vector 2022 skin is the default skin. Half of logged-in desktop users will see an interface where the different parts of the page are more clearly separated. You can [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2023-05 Zebra9 A/B test|read more]]. [https://phabricator.wikimedia.org/T333180][https://phabricator.wikimedia.org/T335972]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] &lt;code&gt;jquery.tipsy&lt;/code&gt; will be removed from the MediaWiki core. This will affect some user scripts. Many lines with &lt;code&gt;.tipsy(&lt;/code&gt; can be commented out. &lt;code&gt;OO.ui.PopupWidget&lt;/code&gt; can be used to keep things working like they are now. You can [[phab:T336019|read more]] and [[:mw:Help:Locating broken scripts|read about how to find broken scripts]]. [https://phabricator.wikimedia.org/T336019]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:36, 9 May 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24998636 --&gt;

== Tech News: 2023-20 ==

&lt;section begin=&quot;technews-2023-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/20|Translations]] are available.

'''Problems'''
* Citations that are automatically generated based on [[d:Q33057|ISBN]] are currently broken. This affects citations made with the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]], and the use of the citoid API in gadgets and user scripts. Work is ongoing to restore this feature. [https://phabricator.wikimedia.org/T336298] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-17|en}}. It will be on all wikis from {{#time:j xg|2023-05-18|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-gorwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hakwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hifwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-htwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-igwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ilowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-inhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308134] 

'''Future changes'''
* There is a recently formed team at the Wikimedia Foundation which will be focusing on experimenting with new tools. Currently they are building [[m:Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI|a prototype ChatGPT plugin that allows information generated by ChatGPT to be properly attributed]] to the Wikimedia projects.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget and userscript developers should replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mediawiki.cookie&lt;/code&gt;&lt;/bdi&gt;. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; library will be removed in ~1 month, and staff developers will run a script to replace any remaining uses at that time. [https://phabricator.wikimedia.org/T336018]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:45, 15 May 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25011501 --&gt;

== Tech News: 2023-21 ==

&lt;section begin=&quot;technews-2023-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/21|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The &quot;recent edits&quot; time period for page watchers is now 30 days. It used to be 180 days. This was a [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Change information about the number of watchers on a page|Community Wishlist Survey proposal]]. [https://phabricator.wikimedia.org/T336250]

'''Changes later this week'''
* An [[mw:special:MyLanguage/Growth/Positive reinforcement#Impact|improved impact module]] will be available at Wikipedias. The impact module is a feature available to newcomers [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer homepage|at their personal homepage]]. It will show their number of edits, how many readers their edited pages have, how many thanks they have received and similar things. It is also accessible by accessing Special:Impact. [https://phabricator.wikimedia.org/T336203]
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-24|en}}. It will be on all wikis from {{#time:j xg|2023-05-25|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W21&quot;/&gt;

16:55, 22 May 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25028325 --&gt;

== Tech News: 2023-22 ==

&lt;section begin=&quot;technews-2023-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/22|Translations]] are available.

'''Recent changes'''
* Citations can once again be added automatically from ISBNs, thanks to Zotero's ISBN searches. The current data sources are the Library of Congress (United States), the Bibliothèque nationale de France (French National Library), and K10plus ISBN (German repository). Additional data source searches can be [[mw:Citoid/Creating Zotero translators|proposed to Zotero]]. The ISBN labels in the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]] will reappear later this week. [https://phabricator.wikimedia.org/T336298#8859917]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The page [[{{#special:EditWatchlist}}]] now has &quot;{{int:watchlistedit-normal-check-all}}&quot; options to select all the pages within a namespace. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Watchlist edit - &quot;check all&quot; checkbox|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334252]	

'''Problems'''
* For a few days earlier this month, the &quot;Add interlanguage link&quot; item in the Tools menu did not work properly. This has now been fixed. [https://phabricator.wikimedia.org/T337081]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-31|en}}. It will be on all wikis from {{#time:j xg|2023-06-01|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* VisualEditor will be switched to a new backend on [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/small.dblist small] and [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/medium.dblist medium] wikis this week. Large wikis will follow in the coming weeks. This is part of the effort to move Parsoid into MediaWiki core. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:03, 29 May 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25079963 --&gt;

== Tech News: 2023-23 ==

&lt;section begin=&quot;technews-2023-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/23|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Help:Extension:RealMe|RealMe]] extension allows you to mark URLs on your user page as verified for Mastodon and similar software.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] Citation and footnote editing can now be started from the reference list when using the visual editor. This feature request was [[m:Community Wishlist Survey 2023/Citations/Allow citations to be edited in the references section with VisualEditor|voted #2 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T54750]
* Previously, clicking on someone else's link to Recent Changes with filters applied within the URL could unintentionally change your preference for &quot;{{int:Rcfilters-group-results-by-page}}&quot;. This has now been fixed. [https://phabricator.wikimedia.org/T202916#8874081]

'''Problems'''
* For a few days last week, some tools and bots returned outdated information due to database replication problems, and may have been down entirely while it was being fixed. These issues have now been fixed. [https://phabricator.wikimedia.org/T337446]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-07|en}}. It will be on all wikis from {{#time:j xg|2023-06-08|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Bots will no longer be prevented from making edits because of URLs that match the [[mw:Special:MyLanguage/Extension:SpamBlacklist|spam blacklist]]. [https://phabricator.wikimedia.org/T313107]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:52, 5 June 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25114640 --&gt;

== Tech News: 2023-24 ==

&lt;section begin=&quot;technews-2023-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/24|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the Dutch, German, Hungarian, Indonesian, Japanese, Polish and Portuguese Wikipedias. This was the [[m:Community Wishlist Survey 2023/Reading/Extend &quot;Who Wrote That?&quot; tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334891]
* The [[mw:Special:MyLanguage/Structured Data Across Wikimedia/Search Improvements#Search Preview panel|Search Preview panel]] has been deployed on four Wikipedias (Catalan, Dutch, Hungarian and Norwegian). The panel will show an image related to the article (if existing), the top sections of the article, related images (coming from MediaSearch on Commons), and eventually the sister projects associated with the article. [https://phabricator.wikimedia.org/T306341]
* The [[:mw:Special:MyLanguage/Help:Extension:RealMe#Verifying_a_link_on_non-user_pages|RealMe]] extension now allows administrators to verify URLs for any page, for Mastodon and similar software. [https://phabricator.wikimedia.org/T324937]
* The default project license [https://lists.wikimedia.org/hyperkitty/list/wikimediaannounce-l@lists.wikimedia.org/thread/7G6XPWZPQFLZ2JANN3ZX6RT4DVUI3HZQ/ has been officially upgraded] to CC BY-SA 4.0. The software interface messages have been updated. Communities should feel free to start updating any mentions of the old CC BY-SA 3.0 licensing within policies and related documentation pages. [https://phabricator.wikimedia.org/T319064]

'''Problems'''
* For three days last month, some Wikipedia pages edited with VisualEditor or DiscussionTools had an unintended &lt;code&gt;&lt;nowiki&gt;__TOC__&lt;/nowiki&gt;&lt;/code&gt; (or its localized form) added during an edit. There is [[mw:Parsoid/Deployments/T336101_followup|a listing of affected pages sorted by wiki]], that may still need to be fixed. [https://phabricator.wikimedia.org/T336101]
* Currently, the &quot;{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}&quot; feature in VisualEditor is broken. Existing &lt;code&gt;&lt;nowiki&gt;{{DEFAULTSORT:...}}&lt;/nowiki&gt;&lt;/code&gt; keywords incorrectly appear as missing templates in VisualEditor. Developers are exploring how to fix this. In the meantime, those wishing to edit the default sortkey of a page are advised to switch to source editing. [https://phabricator.wikimedia.org/T337398]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Last week, an update to the delete form may have broken some gadgets or user scripts. If you need to manipulate (empty) the reason field, replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;#wpReason&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot; style=&quot;white-space: nowrap;&quot;&gt;&lt;code&gt;#wpReason &gt; input&lt;/code&gt;&lt;/bdi&gt;. See [https://cs.wikipedia.org/w/index.php?title=MediaWiki%3AGadget-CleanDeleteReasons.js&amp;diff=22859956&amp;oldid=12794189 an example fix]. [https://phabricator.wikimedia.org/T337809]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-14|en}}. It will be on all wikis from {{#time:j xg|2023-06-15|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* VisualEditor will be switched to a new backend on English Wikipedia on Monday, and all other [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/large.dblist large] wikis on Thursday. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529]

'''Future changes'''
* From 5 June to 17 July, the Foundation's [[:mw:Wikimedia Security Team|Security team]] is holding a consultation with contributors regarding a draft policy to govern the use of third-party resources in volunteer-developed gadgets and scripts. Feedback and suggestions are warmly welcome at [[m:Special:MyLanguage/Third-party resources policy|Third-party resources policy]] on meta-wiki.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:51, 12 June 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25133779 --&gt;

== Tech News: 2023-25 ==

&lt;section begin=&quot;technews-2023-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/25|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Flame graphs are now available in WikimediaDebug. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/JXNQD3EHG5V5QW5UXFDPSHQG4MJ3FWJQ/][https://techblog.wikimedia.org/2023/06/08/flame-graphs-arrive-in-wikimediadebug/]

'''Changes later this week'''
* There is no new MediaWiki version this week.
* There is now a toolbar search popup in the visual editor. You can trigger it by typing &lt;code&gt;\&lt;/code&gt; or pressing &lt;code&gt;ctrl + shift + p&lt;/code&gt;. It can help you quickly access most tools in the editor. [https://commons.wikimedia.org/wiki/File:Visual_editor_toolbar_search_feature.png][https://phabricator.wikimedia.org/T66905]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:08, 19 June 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25159510 --&gt;

== Tech News: 2023-26 ==

&lt;section begin=&quot;technews-2023-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/26|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Action API modules and Special:LinkSearch will now add a trailing &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;/&lt;/code&gt;&lt;/bdi&gt; to all &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prop=extlinks&lt;/code&gt;&lt;/bdi&gt; responses for bare domains. This is part of the work to remove duplication in the &lt;code&gt;externallinks&lt;/code&gt; database table. [https://phabricator.wikimedia.org/T337994]

'''Problems'''
* Last week, search was broken on Commons and Wikidata for 23 hours. [https://phabricator.wikimedia.org/T339810][https://wikitech.wikimedia.org/wiki/Incidents/2023-06-18_search_broken_on_wikidata_and_commons]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-28|en}}. It will be on all wikis from {{#time:j xg|2023-06-29|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Minerva skin now applies more predefined styles to the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.mbox-text&lt;/code&gt;&lt;/bdi&gt; CSS class. This enables support for mbox templates that use divs instead of tables. Please make sure that the new styles won't affect other templates in your wiki. [https://gerrit.wikimedia.org/r/c/mediawiki/skins/MinervaNeue/+/930901/][https://phabricator.wikimedia.org/T339040]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadgets will now load on both desktop and mobile by default. Previously, gadgets loaded only on desktop by default. Changing this default using the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|targets=&lt;/code&gt;&lt;/bdi&gt; parameter is also deprecated and should not be used. You should make gadgets work on mobile or disable them based on the skin (with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|skins=&lt;/code&gt;&lt;/bdi&gt; parameter in &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;MediaWiki:Gadgets-definition&lt;/bdi&gt;) rather than whether the user uses the mobile or the desktop website. Popular gadgets that create errors on mobile will be disabled by developers on the Minerva skin as a temporary solution. [https://phabricator.wikimedia.org/T127268]
* All namespace tabs now have the same browser [[m:Special:MyLanguage/Help:Keyboard_shortcuts|access key]] by default. Previously, custom and extension-defined namespaces would have to have their access keys set manually on-wiki, but that is no longer necessary. [https://phabricator.wikimedia.org/T22126]
* The review form of the Flagged Revisions extension now uses the standardized [[mw:Special:MyLanguage/Codex|user interface components]]. [https://phabricator.wikimedia.org/T191156]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] How media is structured in the parser's HTML output will change in the coming weeks at [[:wikitech:Deployments/Train#Thursday|group2 wikis]]. This change improves the accessibility of content. You may need to update your site-CSS, or userscripts and gadgets. There are [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Media_structure/FAQ|details on what code to check, how to update the code, and where to report any related problems]]. [https://phabricator.wikimedia.org/T314318]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:18, 26 June 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25202311 --&gt;

== Tech News: 2023-27 ==

&lt;section begin=&quot;technews-2023-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/27|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the rolling out of the [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|audio links that play on click]] wishlist proposal, [https://noc.wikimedia.org/conf/highlight.php?file=dblists/small.dblist small wikis] will now be able to use the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline audio player mode|inline audio player]] that is implemented by the [[mw:Extension:Phonos|Phonos]] extension. [https://phabricator.wikimedia.org/T336763]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] From this week all gadgets automatically load on mobile and desktop sites. If you see any problems with gadgets on your wikis, please adjust the [[mw:Special:MyLanguage/Extension:Gadgets#Options|gadget options]] in your gadget definitions file. [https://phabricator.wikimedia.org/T328610]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-05|en}}. It will be on all wikis from {{#time:j xg|2023-07-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:51, 3 July 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25231546 --&gt;

== Tech News: 2023-28 ==

&lt;section begin=&quot;technews-2023-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/28|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Structured Data Across Wikimedia/Section-level Image Suggestions|Section-level Image Suggestions feature]] has been deployed on seven Wikipedias (Portuguese, Russian, Indonesian, Catalan, Hungarian, Finnish and Norwegian Bokmål). The feature recommends images for articles on contributors' watchlists that are a good match for individual sections of those articles.
* [[:m:Special:MyLanguage/Global AbuseFilter|Global abuse filters]] have been enabled on all Wikimedia projects, except English and Japanese Wikipedias (who opted out). This change was made following a [[:m:Requests for comment/Make global abuse filters opt-out|global request for comments]]. [https://phabricator.wikimedia.org/T341159]
* [[{{#special:BlockedExternalDomains}}]] is a new tool for administrators to help fight spam. It provides a clearer interface for blocking plain domains (and their subdomains), is more easily searchable, and is faster for the software to process for each edit on the wiki. It does not support regex (for complex cases), nor URL path-matching, nor the [[MediaWiki:Spam-whitelist|MediaWiki:Spam-whitelist]], but otherwise it replaces most of the functionalities of the existing [[MediaWiki:Spam-blacklist|MediaWiki:Spam-blacklist]]. There is a Python script to help migrate all simple domains into this tool, and more feature details, within [[mw:Special:MyLanguage/Manual:BlockedExternalDomains|the tool's documentation]]. It is available at all wikis except for Meta-wiki, Commons, and Wikidata. [https://phabricator.wikimedia.org/T337431]
* The WikiEditor extension was updated. It includes some of the most frequently used features of wikitext editing. In the past, many of its messages could only be translated by administrators, but now all regular translators on translatewiki can translate them. Please check [https://translatewiki.net/wiki/Special:MessageGroupStats?group=ext-wikieditor&amp;messages=&amp;x=D#sortable:0=asc the state of WikiEditor localization into your language], and if the &quot;Completion&quot; for your language shows anything less than 100%, please complete the translation. See [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/thread/D4YELU2DXMZ75PGELUOKXXMFF3FH45XA/ a more detailed explanation].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-12|en}}. It will be on all wikis from {{#time:j xg|2023-07-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* The default protocol of [[{{#special:LinkSearch}}]] and API counterparts has changed from http to both http and https. [https://phabricator.wikimedia.org/T14810]
* [[{{#special:LinkSearch}}]] and its API counterparts will now search for all of the URL provided in the query. It used to be only the first 60 characters. This feature was requested fifteen years ago. [https://phabricator.wikimedia.org/T17218]

'''Future changes'''
* There is an experiment with a [[:w:en:ChatGPT|ChatGPT]] plugin. This is to show users where the information is coming from when they read information from Wikipedia. It has been tested by Wikimedia Foundation staff and other Wikimedians. Soon all ChatGPT plugin users can use the Wikipedia plugin. This is the same plugin which was mentioned in [[m:Special:MyLanguage/Tech/News/2023/20|Tech News 2023/20]]. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI]
* There is an ongoing discussion on a [[m:Special:MyLanguage/Third-party resources policy|proposed Third-party resources policy]]. The proposal will impact the use of third-party resources in gadgets and userscripts. Based on the ideas received so far, policy includes some of the risks related to user scripts and gadgets loading third-party resources, some best practices and exemption requirements such as code transparency and inspectability. Your feedback and suggestions are warmly welcome until July 17, 2023 on [[m:Talk:Third-party resources policy|on the policy talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:54, 10 July 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25278797 --&gt;

== Tech News: 2023-29 ==

&lt;section begin=&quot;technews-2023-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/29|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] We are now serving 1% of all global user traffic from [[w:en:Kubernetes|Kubernetes]] (you can [[wikitech:MediaWiki On Kubernetes|read more technical details]]). We are planning to increment this percentage regularly. You can [[phab:T290536|follow the progress of this work]]. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-19|en}}. It will be on all wikis from {{#time:j xg|2023-07-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki [[mw:Special:MyLanguage/Help:System_message|system messages]] will now look for available local fallbacks, instead of always using the default fallback defined by software. This means wikis no longer need to override each language on the [[mw:Special:MyLanguage/Manual:Language#Fallback_languages|fallback chain]] separately. For example, English Wikipedia doesn't have to create &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;en-ca&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;en-gb&lt;/code&gt;&lt;/bdi&gt; subpages with a transclusion of the base pages anymore. This makes it easier to maintain local overrides. [https://phabricator.wikimedia.org/T229992]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;action=growthsetmentorstatus&lt;/code&gt;&lt;/bdi&gt; API will be deprecated with the new MediaWiki version. Bots or scripts calling that API should use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;action=growthmanagementorlist&lt;/code&gt;&lt;/bdi&gt; API now. [https://phabricator.wikimedia.org/T321503]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:08, 17 July 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25289122 --&gt;

== Tech News: 2023-30 ==

&lt;section begin=&quot;technews-2023-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/30|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]]   On July 18, the Wikimedia Foundation launched a survey about the [[:mw:Technical_decision_making|technical decision making process]] for people who do technical work that relies on software that is maintained by the Foundation or affiliates. If this applies to you, [https://wikimediafoundation.limesurvey.net/885471 please take part in the survey]. The survey will be open for three weeks, until August 7. You can find more information in [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/Q7DUCFA75DXG3G2KHTO7CEWMLCYTSDB2/|the announcement e-mail on wikitech-l]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-26|en}}. It will be on all wikis from {{#time:j xg|2023-07-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W30&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:20, 25 July 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25332248 --&gt;

== Tech News: 2023-31 ==

&lt;section begin=&quot;technews-2023-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/31|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Synchronizer|Synchronizer]] tool is now available to keep Lua modules synced across Wikimedia wikis, along with [[mw:Multilingual Templates and Modules|updated documentation]] to develop global Lua modules and templates.
* The tag filter on [[{{#special:NewPages}}]] and revision history pages can now be inverted. For example, you can hide edits that were made using an automated tool. [https://phabricator.wikimedia.org/T334337][https://phabricator.wikimedia.org/T334338]
* The Wikipedia [[:w:en:ChatGPT|ChatGPT]] plugin experiment can now be used by ChatGPT users who can use plugins. You can participate in a [[:m:Talk:Wikimedia Foundation Annual Plan/2023-2024/Draft/Future Audiences#Announcing monthly Future Audiences open &quot;office hours&quot;|video call]] if you want to talk about this experiment or similar work. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI]

'''Problems'''
* It was not possible to generate a PDF for pages with non-Latin characters in the title, for the last two weeks. This has now been fixed. [https://phabricator.wikimedia.org/T342442]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-02|en}}. It will be on all wikis from {{#time:j xg|2023-08-03|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Tuesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-kawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kaawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kabwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-knwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kshwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kwwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308135] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:54, 31 July 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25362228 --&gt;

== Tech News: 2023-32 ==

&lt;section begin=&quot;technews-2023-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/32|Translations]] are available.

'''Recent changes'''
* Mobile Web editors can now [[mw:Special:MyLanguage/Reading/Web/Advanced_mobile_contributions#August_1,_2023_-_Full-page_editing_added_on_mobile|edit a whole page at once]]. To use this feature, turn on &quot;{{int:Mobile-frontend-mobile-option-amc}}&quot; in your settings and use the &quot;{{int:Minerva-page-actions-editfull}}&quot; button in the &quot;{{int:Minerva-page-actions-overflow}}&quot; menu. [https://phabricator.wikimedia.org/T203151]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:20, 7 August 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25420038 --&gt;

== Tech News: 2023-33 ==

&lt;section begin=&quot;technews-2023-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/33|Translations]] are available.

'''Recent changes'''
* The Content translation system is no longer using Youdao's [[mw:Special:MyLanguage/Help:Content_translation/Translating/Initial_machine_translation|machine translation service]]. The service was in place for several years, but due to no usage, and availability of alternatives, it was deprecated to reduce maintenance overheads. Other services which cover the same languages are still available. [https://phabricator.wikimedia.org/T329137]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-16|en}}. It will be on all wikis from {{#time:j xg|2023-08-17|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-lawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ladwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lezwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lfnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-liwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lijwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lmowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ltgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-maiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-map_bmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mdfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kywiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308136] &lt;!-- TODO replace wiki codes --&gt;

'''Future changes'''
* A few gadgets/user scripts which add icons to the Minerva skin need to have their CSS updated. There are more details available including a [[phab:T344067|search for all existing instances and how to update them]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 05:59, 15 August 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25428668 --&gt;

== Tech News: 2023-34 ==

&lt;section begin=&quot;technews-2023-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/34|Translations]] are available.

'''Recent changes'''
* The [https://gdrive-to-commons.toolforge.org/ GDrive to Commons Uploader] tool is now available. It enables [[m:Special:MyLanguage/GDrive to Commons Uploader|securely selecting and uploading files]] from your Google Drive directly to Wikimedia Commons. [https://phabricator.wikimedia.org/T267868]
* From now on, we will announce new Wikimedia wikis in Tech News, so you can update any tools or pages.
** Since the last edition, two new wikis have been created:
*** a Wiktionary in [[d:Q7121294|Pa'O]] ([[wikt:blk:|&lt;code&gt;wikt:blk:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T343540]
*** a Wikisource in [[d:Q34002|Sundanese]] ([[s:su:|&lt;code&gt;s:su:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T343539]
** To catch up, the next most recent six wikis are:
*** Wikifunctions ([[f:|&lt;code&gt;f:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T275945]
*** a Wiktionary in [[d:Q2891049|Mandailing]] ([[wikt:btm:|&lt;code&gt;wikt:btm:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335216]
*** a Wikipedia in [[d:Q5555465|Ghanaian Pidgin]] ([[w:gpe:|&lt;code&gt;w:gpe:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335969]
*** a Wikinews in [[d:Q3111668|Gungbe]] ([[n:guw:|&lt;code&gt;n:guw:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T334394]
*** a Wiktionary in [[d:Q33522|Kabardian]] ([[wikt:kbd:|&lt;code&gt;wikt:kbd:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T333266]
*** a Wikipedia in [[d:Q35570|Fante]] ([[w:fat:|&lt;code&gt;w:fat:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335016]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-23|en}}. It will be on all wikis from {{#time:j xg|2023-08-24|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There is an existing [[mw:Stable interface policy|stable interface policy]] for MediaWiki backend code. There is a [[mw:User:Jdlrobson/Stable interface policy/frontend|proposed stable interface policy for frontend code]]. This is relevant for anyone who works on gadgets or Wikimedia frontend code. You can read it, discuss it, and let the proposer know if there are any problems. [https://phabricator.wikimedia.org/T344079]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W34&quot;/&gt;

15:25, 21 August 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25497111 --&gt;

== Tech News: 2023-35 ==

&lt;section begin=&quot;technews-2023-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/35|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Community Wishlist Survey 2022/Better diff handling of paragraph splits|better diff handling of paragraph splits]], improved detection of splits is being rolled out. Over the last two weeks, we deployed this support to [[wikitech:Deployments/Train#Groups|group0]] and group1 wikis. This week it will be deployed to group2 wikis. [https://phabricator.wikimedia.org/T341754]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] All [[{{#special:Contributions}}]] pages now show the user's local edit count and the account's creation date. [https://phabricator.wikimedia.org/T324166]
* Wikisource users can now use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prpbengalicurrency&lt;/code&gt;&lt;/bdi&gt; label to denote Bengali currency characters as page numbers inside the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;pagelist&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; tag.  [https://phabricator.wikimedia.org/T268932]
* Two preferences have been relocated. The preference &quot;{{int:visualeditor-preference-visualeditor}}&quot; is now shown on the [[Special:Preferences#mw-prefsection-editing|&quot;{{int:prefs-editing}}&quot; tab]] at all wikis. Previously it was shown on the &quot;{{int:prefs-betafeatures}}&quot; tab at some wikis. The preference &quot;{{int:visualeditor-preference-newwikitexteditor-enable}}&quot; is now also shown on the &quot;{{int:prefs-editing}}&quot; tab at all wikis, instead of the &quot;{{int:prefs-betafeatures}}&quot; tab. [https://phabricator.wikimedia.org/T335056][https://phabricator.wikimedia.org/T344158]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-30|en}}. It will be on all wikis from {{#time:j xg|2023-08-31|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] New signups for a Wikimedia developer account will start being pushed towards &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://idm.wikimedia.org/ idm.wikimedia.org]&lt;/bdi&gt;, rather than going via Wikitech. [[wikitech:IDM|Further information about the new system is available]].
* All right-to-left language wikis, plus Korean, Armenian, Ukrainian, Russian, and Bulgarian Wikipedias, will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. This feature will come to more wikis in future weeks. [https://phabricator.wikimedia.org/T267921]

'''Future changes'''
* The removal of the [[mw:Special:MyLanguage/Extension:DoubleWiki|DoubleWiki extension]] is being discussed. This extension currently allows Wikisource users to view articles from multiple language versions side by side when the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;=&gt;&lt;/code&gt;&lt;/bdi&gt; symbol next to a specific language edition is selected. Comments on this are welcomed at [[phab:T344544|the phabricator task]].
* A proposal has been made to merge the second hidden-categories list (which appears below the wikitext editing form) with the main list of categories (which is further down the page). [[phab:T340606|More information is available on Phabricator]]; feedback is welcome!

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:00, 28 August 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25510866 --&gt;

== Tech News: 2023-36 ==

&lt;section begin=&quot;technews-2023-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/36|Translations]] are available.

'''Recent changes'''
* [[m:Wikisource_EditInSequence|EditInSequence]], a feature that allows users to edit pages faster on Wikisource has been moved to a Beta Feature based on community feedback. To enable it, you can navigate to the [[Special:Preferences#mw-prefsection-betafeatures|beta features tab in Preferences]]. [https://phabricator.wikimedia.org/T308098]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Generate Audio for IPA|Generate Audio for IPA]] and [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|Audio links that play on click]] wishlist proposals, the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline_audio_player_mode|inline audio player mode]] of [[mw:Extension:Phonos|Phonos]] has been deployed to all projects. [https://phabricator.wikimedia.org/T336763]
* There is a new option for Administrators when they are changing the usergroups for a user, to add the user’s user page to their watchlist. This works both via [[{{#special:UserRights}}]] and via the API. [https://phabricator.wikimedia.org/T272294]
* One new wiki has been created:
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q34318|Talysh]] ([[w:tly:|&lt;code&gt;w:tly:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T345166]
'''Problems'''
* The [[mw:Special:MyLanguage/Extension:LoginNotify|LoginNotify extension]] was not sending notifications since January. It has now been fixed, so going forward, you may see notifications for failed login attempts, and successful login attempts from a new device. [https://phabricator.wikimedia.org/T344785]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-06|en}}. It will be on all wikis from {{#time:j xg|2023-09-07|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-mhrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-miwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-minwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mtwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mwlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-myvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mznwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-napwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ndswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nds_nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-novwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nqowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nrmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nsowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ocwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-olowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-omwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-orwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-oswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pagwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-papwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pcdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pdcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pflwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pihwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pnbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pntwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pswiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308137][https://phabricator.wikimedia.org/T308138]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:33, 4 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25566983 --&gt;

== Tech News: 2023-37 ==

&lt;section begin=&quot;technews-2023-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/37|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/ORES|ORES]], the revision evaluation service, is now using a new open-source infrastructure on all wikis except for English Wikipedia and Wikidata. These two will follow this week. If you notice any unusual results from the Recent Changes filters that are related to ORES (for example, &quot;{{int:ores-rcfilters-damaging-title}}&quot; and &quot;{{int:ores-rcfilters-goodfaith-title}}&quot;), please [[mw:Talk:Machine Learning|report them]]. [https://phabricator.wikimedia.org/T342115]
* When you are logged in on one Wikimedia wiki and visit a different Wikimedia wiki, the system tries to log you in there automatically. This has been unreliable for a long time. You can now visit the login page to make the system try extra hard. If you feel that made logging in better or worse than it used to be, your feedback is appreciated. [https://phabricator.wikimedia.org/T326281]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-13|en}}. It will be on all wikis from {{#time:j xg|2023-09-14|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/Technical decision making|Technical Decision-Making Forum Retrospective]] team invites anyone involved in the technical field of Wikimedia projects to signup to and join [[mw:Technical decision making/Listening Sessions|one of their listening sessions]] on 13 September. Another date will be scheduled later. The goal is to improve the technical decision-making processes.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Better diff handling of paragraph splits|Better diff handling of paragraph splits]] wishlist proposal, the inline switch widget in diff pages is being rolled out this week to all wikis. The inline switch will allow viewers to toggle between a unified inline or two-column diff wikitext format. [https://phabricator.wikimedia.org/T336716]

'''Future changes'''
* All wikis will be read-only for a few minutes on 20 September. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T345263]
* The Enterprise API is launching a new feature called &quot;[http://breakingnews-beta.enterprise.wikimedia.com/ breaking news]&quot;. Currently in BETA, this attempts to identify likely &quot;newsworthy&quot; topics as they are currently being written about in any Wikipedia. Your help is requested to improve the accuracy of its detection model, especially on smaller language editions, by recommending templates or identifiable editing patterns. See more information at [[mw:Special:MyLanguage/Wikimedia Enterprise/Breaking news|the documentation page]] on MediaWiki or [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#What is Breaking News|the FAQ]] on Meta.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:07, 11 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25589064 --&gt;

== Tech News: 2023-38 ==

&lt;section begin=&quot;technews-2023-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/38|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki now has a [[mw:Stable interface policy/frontend|stable interface policy for frontend code]] that more clearly defines how we deprecate MediaWiki code and wiki-based code (e.g. gadgets and user scripts). Thank you to everyone who contributed to the content and discussions. [https://phabricator.wikimedia.org/T346467][https://phabricator.wikimedia.org/T344079]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-20|en}}. It will be on all wikis from {{#time:j xg|2023-09-21|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* All wikis will be read-only for a few minutes on September 20. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] [https://phabricator.wikimedia.org/T345263]
* All wikis will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. [https://phabricator.wikimedia.org/T267921]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The team investigating the Graph Extension posted [[mw:Extension:Graph/Plans#Proposal|a proposal for reenabling it]] and they need your input.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:19, 18 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25623533 --&gt;

== Tech News: 2023-39 ==

&lt;section begin=&quot;technews-2023-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/39|Translations]] are available.

'''Recent changes'''
* The Vector 2022 skin will now remember the pinned/unpinned status for the Table of Contents for all logged-out users. [https://phabricator.wikimedia.org/T316060]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-27|en}}. It will be on all wikis from {{#time:j xg|2023-09-28|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The ResourceLoader &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mediawiki.ui&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; modules are now deprecated as part of the move to Vue.js and Codex. There is a [[mw:Codex/Migrating_from_MediaWiki_UI|guide for migrating from MediaWiki UI to Codex]] for any tools that use it. More [[phab:T346468|details are available in the task]] and your questions are welcome there.
* Gadget definitions will have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new &quot;namespaces&quot; option]]. The option takes a list of namespace IDs. Gadgets that use this option will only load on pages in the given namespaces.

'''Future changes'''
* New variables will be added to [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]]: &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;global_account_groups&lt;/bdi&gt;&lt;/code&gt; and &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;global_account_editcount&lt;/bdi&gt;&lt;/code&gt;. They are available only when an account is being created. You can use them to prevent blocking automatic creation of accounts when users with many edits elsewhere visit your wiki for the first time. [https://phabricator.wikimedia.org/T345632][https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:AbuseFilter/Rules_format]

'''Meetings'''
* You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1698426015 27 October at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia_Apps/Office_Hours#October_2023|details and how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:51, 26 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25655264 --&gt;

== Tech News: 2023-40 ==

&lt;section begin=&quot;technews-2023-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/40|Translations]] are available.

'''Recent changes'''
* There is a new [[Special:Preferences#mw-prefsection-rendering-advancedrendering|user preference]] for &quot;{{int:tog-forcesafemode}}&quot;. This setting will make pages load without including any on-wiki JavaScript or on-wiki stylesheet pages. It can be useful for debugging broken JavaScript gadgets. [https://phabricator.wikimedia.org/T342347]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget definitions now have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new &quot;&lt;var&gt;contentModels&lt;/var&gt;&quot; option]]. The option takes a list of page content models, like &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;wikitext&lt;/bdi&gt;&lt;/code&gt; or &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;css&lt;/bdi&gt;&lt;/code&gt;. Gadgets that use this option will only load on pages with the given content models.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-04|en}}. It will be on all wikis from {{#time:j xg|2023-10-05|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Vector 2022 skin will no longer use the custom styles and scripts of Vector legacy (2010). The change will be made later this year or in early 2024. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|how to adjust the CSS and JS pages on your wiki]]. [https://phabricator.wikimedia.org/T331679]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:26, 3 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25686930 --&gt;

== Tech News: 2023-41 ==

&lt;section begin=&quot;technews-2023-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/41|Translations]] are available.

'''Recent changes'''
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q33291|Fon]] ([[w:fon:|&lt;code&gt;w:fon:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T347935]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-11|en}}. It will be on all wikis from {{#time:j xg|2023-10-12|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-swwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-warwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xalwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xmfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zeawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zh_min_nanwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zuwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308139]
* At some wikis, newcomers are suggested images from Commons to add to articles without any images. Starting on Tuesday, newcomers at these wikis will be able to add images to unillustrated article sections. The specific wikis are listed under &quot;Images recommendations&quot; [[mw:Special:MyLanguage/Growth/Deployment table|at the Growth team deployment table]]. You can [[mw:Special:MyLanguage/Help:Growth/Tools/Add an image|learn more about this feature.]] [https://phabricator.wikimedia.org/T345940]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In the mobile web skin (Minerva) the CSS ID &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#page-actions&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will be replaced with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#p-views&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This change is to make it consistent with other skins and to improve support for gadgets and extensions in the mobile skin. A few gadgets may need to be updated; there are [https://phabricator.wikimedia.org/T348267 details and search-links in the task].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W41&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:39, 9 October 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25712895 --&gt;

== Tech News: 2023-42 ==

&lt;section begin=&quot;technews-2023-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/42|Translations]] are available.

'''Recent changes'''
* The [[m:Special:MyLanguage/Help:Unified login|Unified login]] system's edge login should now be fixed for some browsers (Chrome, Edge, Opera). This means that if you visit a new sister project wiki, you should be logged in automatically without the need to click &quot;Log in&quot; or reload the page. Feedback on whether it's working for you is welcome. [https://phabricator.wikimedia.org/T347889]
* [[mw:Special:MyLanguage/Manual:Interface/Edit_notice|Edit notices]] are now available within the MobileFrontend/Minerva skin. This feature was inspired by [[w:en:Wikipedia:EditNoticesOnMobile|the gadget on English Wikipedia]]. See more details in [[phab:T316178|T316178]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-18|en}}. It will be on all wikis from {{#time:j xg|2023-10-19|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* In 3 weeks, in the Vector 2022 skin, code related to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;addPortletLink&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#p-namespaces&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; that was deprecated one year ago will be removed. If you notice tools that should appear next to the &quot;Discussion&quot; tab are then missing, please tell the gadget's maintainers to see [[phab:T347907|instructions in the Phabricator task]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:47, 16 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25745824 --&gt;

== Tech News: 2023-43 ==

&lt;section begin=&quot;technews-2023-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/43|Translations]] are available.

'''Recent changes'''
* There is a new [[mw:Special:MyLanguage/Wikimedia Language engineering/Newsletter/2023/October|Language and internationalization newsletter]], written quarterly. It contains updates on new feature development, improvements in various language-related technical projects, and related support work.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Source map support has been enabled on all wikis. When you open the debugger in your browser's developer tools, you should be able to see the unminified JavaScript source code. [https://phabricator.wikimedia.org/T47514]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-25|en}}. It will be on all wikis from {{#time:j xg|2023-10-26|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:16, 23 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25782286 --&gt;

== Tech News: 2023-44 ==

&lt;section begin=&quot;technews-2023-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/44|Translations]] are available.

'''Recent changes'''
* The Structured Content team, as part of its project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]], made some UX improvements to the upload step of choosing own vs not own work ([[phab:T347590|T347590]]), as well as to the licensing step for own work ([[phab:T347756|T347756]]).
* The Design Systems team has released version 1.0.0 of [[wmdoc:codex/latest/|Codex]], the new design system for Wikimedia. See the [[mw:Special:MyLanguage/Design_Systems_Team/Announcing_Codex_1.0|full announcement about the release of Codex 1.0.0]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-01|en}}. It will be on all wikis from {{#time:j xg|2023-11-02|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).
* Listings on category pages are sorted on each wiki for that language using a [[:w:en:International Components for Unicode|library]]. For a brief period on 2 November, changes to categories will not be sorted correctly for many languages. This is because the developers are upgrading to a new version of the library. They will then use a script to fix the existing categories. This will take a few hours or a few days depending on how big the wiki is. You can [[mw:Special:MyLanguage/Wikimedia Technical Operations/ICU announcement|read more]]. [https://phabricator.wikimedia.org/T345561][https://phabricator.wikimedia.org/T267145] 
* Starting November 1, the impact module (Special:Impact) will be upgraded by the Growth team. The new impact module shows newcomers more data regarding their impact on the wiki. It was tested by a few wikis during the last few months. [https://phabricator.wikimedia.org/T336203]

'''Future changes'''
* There is [[mw:Special:MyLanguage/Extension:Graph/Plans#Roadmap|a proposed plan]] for re-enabling the Graph Extension. You can help by reviewing this proposal and [[mw:Extension_talk:Graph/Plans#c-PPelberg_(WMF)-20231020221600-Update:_20_October|sharing what you think about it]].
* The WMF is working on making it possible for administrators to [[mw:Special:MyLanguage/Community_configuration_2.0|edit MediaWiki configuration directly]]. This is similar to previous work on Special:EditGrowthConfig. [[phab:T349757|A technical RfC is running until November 08, where you can provide feedback.]] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:21, 30 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25801989 --&gt;

== Tech News: 2023-45 ==

&lt;section begin=&quot;technews-2023-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/45|Translations]] are available.

'''Recent changes'''
* In the Vector 2022 skin, the default font-size of a number of navigational elements (tagline, tools menu, navigational links, and more) has been increased slightly to match the font size used in page content. [https://phabricator.wikimedia.org/T346062]

'''Problems'''
* Last week, there was a problem displaying some recent edits on [https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist a few wikis], for 1-6 hours. The edits were saved but not immediately shown. This was due to a database problem. [https://phabricator.wikimedia.org/T350443]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-08|en}}. It will be on all wikis from {{#time:j xg|2023-11-09|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).
* The Growth team will reassign newcomers from former mentors to [[mw:Special:MyLanguage/Growth/Structured mentor list|the currently active mentors]]. They have also changed the notification language to be more user-friendly. [https://phabricator.wikimedia.org/T330071][https://phabricator.wikimedia.org/T327493]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:05, 6 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25838105 --&gt;

== Tech News: 2023-46 ==

&lt;section begin=&quot;technews-2023-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/46|Translations]] are available.

'''Recent changes'''
* Four new wikis have been created:
** a Wikipedia in [[d:Q7598268|Moroccan Amazigh]] ([[w:zgh:|&lt;code&gt;w:zgh:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350216]
** a Wikipedia in [[d:Q35159|Dagaare]] ([[w:dga:|&lt;code&gt;w:dga:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350218]
** a Wikipedia in [[d:Q33017|Toba Batak]] ([[w:bbc:|&lt;code&gt;w:bbc:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350320]
** a Wikiquote in [[d:Q33151|Banjar]] ([[q:bjn:|&lt;code&gt;q:bjn:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350217]

'''Problems'''
* Last week, users who previously visited Meta-Wiki or Wikimedia Commons and then became logged out on those wikis could not log in again. The problem is now resolved. [https://phabricator.wikimedia.org/T350695]
* Last week, some pop-up dialogs and menus were shown with the wrong font size. The problem is now resolved. [https://phabricator.wikimedia.org/T350544]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-15|en}}. It will be on all wikis from {{#time:j xg|2023-11-16|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).

'''Future changes'''
* Reference Previews are coming to many wikis as a default feature. They are popups for references, similar to the [[mw:Special:MyLanguage/Page Previews|PagePreviews feature]]. [[m:WMDE Technical Wishes/ReferencePreviews#Opt-out feature|You can opt out]] of seeing them. If you are [[Special:Preferences#mw-prefsection-gadgets|using the gadgets]] Reference Tooltips or Navigation Popups, you won’t see Reference Previews. [[phab:T282999|Deployment]] is planned for November 22, 2023. 
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Canary (also known as heartbeat) events will be produced into [https://stream.wikimedia.org/?doc#/streams Wikimedia event streams] from December 11. Streams users are advised to filter out these events, by discarding all events where &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;meta.domain == &quot;canary&quot;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Updates to [[mw:Special:MyLanguage/Manual:Pywikibot|Pywikibot]] or [https://github.com/ChlodAlejandro/wikimedia-streams wikimedia-streams] will discard these events by default. [https://phabricator.wikimedia.org/T266798]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:52, 13 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25859263 --&gt;

== Tech News: 2023-47 ==

&lt;section begin=&quot;technews-2023-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/47|Translations]] are available.

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-quwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_rupwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_tarawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ruewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rwwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-satwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-shwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-siwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-skwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-slwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-smwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-suwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-szlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tcywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tetwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-towiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tpiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ttwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-twwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tyvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-udmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ugwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-uzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vecwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vepwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vlswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vowiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308141][https://phabricator.wikimedia.org/T308142][https://phabricator.wikimedia.org/T308143] 
* The Vector 2022 skin will have some minor visual changes to drop-down menus, column widths, and more. These changes were added to four Wikipedias last week. If no issues are found, these changes will proceed to all wikis this week. These changes will make it possible to add new menus for readability and dark mode. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Updates#November_2023:_Visual_changes,_more_deployments,_and_shifting_focus|Learn more]]. [https://phabricator.wikimedia.org/T347711]

'''Future changes'''
* There is [[mw:Extension talk:Graph/Plans#Update: 15 November|an update on re-enabling the Graph Extension]]. To speed up the process, Vega 2 will not be supported and only [https://phabricator.wikimedia.org/T335325 some protocols] will be available at launch. You can help by sharing what you think about the plan.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:55, 21 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25884616 --&gt;

== Tech News: 2023-48 ==

&lt;section begin=&quot;technews-2023-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/48|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-29|en}}. It will be on all wikis from {{#time:j xg|2023-11-30|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki's JavaScript system will now allow &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;async&lt;/code&gt;/&lt;code&gt;await&lt;/code&gt;&lt;/bdi&gt; syntax in gadgets and user scripts. Gadget authors should remember that users' browsers may not support it, so it should be used appropriately. [https://phabricator.wikimedia.org/T343499]
* The deployment of &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add_a_link|Add a link]]&quot; announced [[m:Special:MyLanguage/Tech/News/2023/47|last week]] was postponed. It will resume this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:08, 27 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25906379 --&gt;

== Tech News: 2023-49 ==

&lt;section begin=&quot;technews-2023-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/49|Translations]] are available.

'''Recent changes'''
* The spacing between paragraphs on Vector 2022 has been changed from 7px to 14px to match the size of the text. This will make it easier to distinguish paragraphs from sentences. [https://phabricator.wikimedia.org/T351754]
* The &quot;{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}&quot; feature in VisualEditor is working again. You no longer need to switch to source editing to edit &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{DEFAULTSORT:...}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; keywords. [https://phabricator.wikimedia.org/T337398]

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* On 6 December, people who have the enabled the preference for &quot;{{int:Discussiontools-preference-visualenhancements}}&quot; will notice the [[mw:Special:MyLanguage/Talk pages project/Usability|talk page usability improvements]] appear on pages that include the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__NEWSECTIONLINK__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; magic word. If you notice any issues, please [[phab:T352232|share them with the team on Phabricator]].

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge [[wikitech:News/Toolforge Grid Engine deprecation|Grid Engine shutdown process]] will start on December 14. Maintainers of [[toolforge:grid-deprecation|tools that still use this old system]] should plan to migrate to Kubernetes, or tell the team your plans on Phabricator in the task about your tool, before that date. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/VIWWQKMSQO2ED3TVUR7KPPWRTOBYBVOA/]
* Communities using [[mw:Special:MyLanguage/Structured_Discussions|Structured Discussions]] are being contacted regarding [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|the upcoming deprecation of Structured Discussions]]. You can read more about this project, and share your comments, [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|on the project's page]].

'''Events'''
* Registration &amp; Scholarship applications are now open for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2024|Wikimedia Hackathon 2024]] that will take place from 3–5 May in Tallinn, Estonia. Scholarship applications are open until 5 January 2024.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:50, 4 December 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25914435 --&gt;

== Tech News: 2023-50 ==

&lt;section begin=&quot;technews-2023-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/50|Translations]] are available.

'''Recent changes'''
* On Wikimedia Commons, there are some minor user-interface improvements for the &quot;choosing own vs not own work&quot; step in the UploadWizard. This is part of the Structured Content team's project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]]. [https://phabricator.wikimedia.org/T352707][https://phabricator.wikimedia.org/T352709]

'''Problems'''
* There was a problem showing the [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer homepage|Newcomer homepage]] feature with the &quot;impact module&quot; and their page-view graphs, for a few days in early December. This has now been fixed. [https://phabricator.wikimedia.org/T352352][https://phabricator.wikimedia.org/T352349]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-13|en}}. It will be on all wikis from {{#time:j xg|2023-12-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=]] The [https://wikimediafoundation.limesurvey.net/796964 2023 Developer Satisfaction Survey] is seeking the opinions of the Wikimedia developer community. Please take the survey if you have any role in developing software for the Wikimedia ecosystem. The survey is open until 5 January 2024, and has an associated [[foundation:Legal:December_2023_Developer_Satisfaction_Survey|privacy statement]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:12, 12 December 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25945501 --&gt;

== Tech News: 2023-51 ==

&lt;section begin=&quot;technews-2023-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/51|Translations]] are available.

'''Tech News'''
* The next issue of Tech News will be sent out on 8 January 2024 because of [[w:en:Christmas and holiday season|the holidays]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-20|en}}. It will be on all wikis from {{#time:j xg|2023-12-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting December 18, it won't be possible to activate Structured Discussions on a user's own talk page using the Beta feature. The Beta feature option remains available for users who want to deactivate Structured Discussions. This is part of [[mw:Structured Discussions/Deprecation|Structured Discussions' deprecation work]]. [https://phabricator.wikimedia.org/T248309] 
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There will be full support for redirects in the Module namespace. The &quot;Move Page&quot; feature will leave an appropriate redirect behind, and such redirects will be appropriately recognized by the software (e.g. hidden from [[{{#special:UnconnectedPages}}]]). There will also be support for [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Renaming or moving modules|manual redirects]]. [https://phabricator.wikimedia.org/T120794]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The MediaWiki JavaScript documentation is moving to a new format. During the move, you can read the old docs using [https://doc.wikimedia.org/mediawiki-core/REL1_41/js/ version 1.41]. Feedback about [https://doc.wikimedia.org/mediawiki-core/master/js/ the new site] is welcome on the [[mw:Talk:JSDoc_WMF_theme|project talk page]].
* The Wishathon is a new initiative that encourages collaboration across the Wikimedia community to develop solutions for wishes collected through the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. The first community Wishathon will take place from 15–17 March. If you are interested in a project proposal as a user, developer, designer, or product lead, you can [[m:Special:MyLanguage/Event:WishathonMarch2024|register for the event and read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:17, 18 December 2023 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25959059 --&gt;

== Tech News: 2024-02 ==

&lt;section begin=&quot;technews-2024-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/02|Translations]] are available.

'''Recent changes'''
* [https://mediawiki2latex.wmflabs.org/ mediawiki2latex] is a tool that converts wiki content into the formats of LaTeX, PDF, ODT, and EPUB. The code now runs many times faster due to recent improvements. There is also an optional Docker container you can [[b:de:Benutzer:Dirk_Hünniger/wb2pdf/install#Using_Docker|install]] on your local machine.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The way that Random pages are selected has been updated. This will slowly reduce the problem of some pages having a lower chance of appearing. [https://phabricator.wikimedia.org/T309477]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-10|en}}. It will be on all wikis from {{#time:j xg|2024-01-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W02&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:19, 9 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26026251 --&gt;

== Tech News: 2024-03 ==

&lt;section begin=&quot;technews-2024-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/03|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Pages that use the JSON [[mw:Special:MyLanguage/Manual:ContentHandler|contentmodel]] will now use tabs instead of spaces for auto-indentation. This will significantly reduce the page size. [https://phabricator.wikimedia.org/T326065]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] and personal user scripts may now use JavaScript syntax introduced in ES6 (also known as &quot;ES2015&quot;) and ES7 (&quot;ES2016&quot;). MediaWiki validates the source code to protect other site functionality from syntax errors, and to ensure scripts are valid in all [[mw:Special:MyLanguage/Compatibility#Browsers|supported browsers]]. Previously, Gadgets could use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;requiresES6&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; option. This option is no longer needed and will be removed in the future. [https://phabricator.wikimedia.org/T75714]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Manual:Bot passwords|Bot passwords]] and [[mw:Special:MyLanguage/OAuth/Owner-only consumers|owner-only OAuth consumers]] can now be restricted to allow editing only specific pages. [https://phabricator.wikimedia.org/T349957]
* You can now [[mw:Special:MyLanguage/Extension:Thanks|thank]] edits made by bots. [https://phabricator.wikimedia.org/T341388]
* An update on the status of the Community Wishlist Survey for 2024 [[m:Special:MyLanguage/Community Wishlist Survey/Future Of The Wishlist/January 4, 2024 Update|has been published]]. Please read and give your feedback.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-17|en}}. It will be on all wikis from {{#time:j xg|2024-01-18|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting on January 17, it will not be possible to login to Wikimedia wikis from some specific old versions of the Chrome browser (versions 51–66, released between 2016 and 2018). Additionally, users of iOS 12, or Safari on Mac OS 10.14, may need to login to each wiki separately. [https://phabricator.wikimedia.org/T344791]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; module was deprecated and replaced with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mediawiki.cookie&lt;/code&gt;&lt;/bdi&gt; module last year. A script has now been run to replace any remaining uses, and this week the temporary alias will be removed. [https://phabricator.wikimedia.org/T354966]

'''Future changes'''
* Wikimedia Deutschland is working to [[m:WMDE Technical Wishes/Reusing references|make reusing references easier]]. They are looking for people who are interested in participating in [https://wikimedia.sslsurvey.de/User-research-into-Reusing-References-Sign-up-Form-2024/en/ individual video calls for user research in January and February].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:13, 16 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26074460 --&gt;

== Tech News: 2024-04 ==

&lt;section begin=&quot;technews-2024-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/04|Translations]] are available.

'''Problems'''
* A bug in UploadWizard prevented linking to the userpage of the uploader when uploading. It has now been fixed. [https://phabricator.wikimedia.org/T354529]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-24|en}}. It will be on all wikis from {{#time:j xg|2024-01-25|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:03, 23 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26096197 --&gt;

== Tech News: 2024-05 ==

&lt;section begin=&quot;technews-2024-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/05|Translations]] are available.

'''Recent changes'''
* Starting Monday January 29, all talk pages messages' timestamps will become a link. This link is a permanent link to the comment. It allows users to find the comment they are looking for, even if this comment was moved elsewhere. This will affect all wikis except for the English Wikipedia. You can read more about this change [https://diff.wikimedia.org/2024/01/29/talk-page-permalinks-dont-lose-your-threads/ on Diff] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk_pages_permalinking|on Mediawiki.org]].&lt;!-- The Diff post will be published on Monday morning UTC--&gt; [https://phabricator.wikimedia.org/T302011]
* There are some improvements to the CAPTCHA to make it harder for spam bots and scripts to bypass it. If you have feedback on this change, please comment on [[phab:T141490|the task]]. Staff are monitoring metrics related to the CAPTCHA, as well as secondary metrics such as account creations and edit counts.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-31|en}}. It will be on all wikis from {{#time:j xg|2024-02-01|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] On February 1, a link will be added to the &quot;Tools&quot; menu to download a [[w:en:QR code|QR code]] that links to the page you are viewing. There will also be a new [[{{#special:QrCode}}]] page to create QR codes for any Wikimedia URL. This addresses the [[m:Community Wishlist Survey 2023/Mobile and apps/Add ability to share QR code for a page in any Wikimedia project|#19 most-voted wish]] from the [[m:Community Wishlist Survey 2023/Results|2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T329973]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] which only work in some skins have sometimes used the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;targets&lt;/code&gt;&lt;/bdi&gt; option to limit where you can use them. This will stop working this week. You should use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;skins&lt;/code&gt;&lt;/bdi&gt; option instead. [https://phabricator.wikimedia.org/T328497]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:31, 29 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26137870 --&gt;

== Tech News: 2024-06 ==

&lt;section begin=&quot;technews-2024-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/06|Translations]] are available.

'''Recent changes'''
*The mobile site history pages now use the same HTML as the desktop history pages. If you hear of any problems relating to mobile history usage please point them to [[phab:T353388|the phabricator task]].
*On most wikis, admins can now block users from making specific actions. These actions are: uploading files, creating new pages, moving (renaming) pages, and sending thanks. The goal of this feature is to allow admins to apply blocks that are adequate to the blocked users' activity. [[m:Special:MyLanguage/Community health initiative/Partial blocks#action-blocks|Learn more about &quot;action blocks&quot;]]. [https://phabricator.wikimedia.org/T242541][https://phabricator.wikimedia.org/T280531]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-07|en}}. It will be on all wikis from {{#time:j xg|2024-02-08|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Talk pages permalinks that included diacritics and non-Latin script were malfunctioning. This issue is fixed. [https://phabricator.wikimedia.org/T356199]

'''Future changes'''
* [[m:WMDE Technical Wishes/ReferencePreviews#24WPs|24 Wikipedias]] with [[mw:Special:MyLanguage/Reference_Tooltips|Reference Tooltips]] as a default gadget are encouraged to remove that default flag. This would make [[mw:Special:MyLanguage/Help:Reference_Previews|Reference Previews]] the new default for reference popups, leading to a more consistent experience across wikis. For [[m:WMDE Technical Wishes/ReferencePreviews#46WPs|46 Wikipedias]] with less than 4 interface admins, the change is already scheduled for mid-February, [[m:Talk:WMDE Technical Wishes/ReferencePreviews#Reference Previews to become the default for previewing references on more wikis.|unless there are concerns]]. The older Reference Tooltips gadget will still remain usable and will override this feature, if it is available on your wiki and you have enabled it in your settings. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/ReferencePreviews#Reference_Previews_to_become_the_default_for_previewing_references_on_more_wikis][https://phabricator.wikimedia.org/T355312]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:22, 5 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26180971 --&gt;

== Tech News: 2024-07 ==

&lt;section begin=&quot;technews-2024-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/07|Translations]] are available.

'''Recent changes'''
*  The [[d:Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split experiment]] is working and loaded onto 3 test servers. The team in charge is testing the split's impact and requires feedback from WDQS users through the UI or programmatically in different channels. [https://www.wikidata.org/wiki/Wikidata_talk:SPARQL_query_service/WDQS_graph_split][https://phabricator.wikimedia.org/T356773][https://www.wikidata.org/wiki/User:Sannita_(WMF)] Users' feedback will validate the impact of various use cases and workflows around the Wikidata Query service. [https://www.wikidata.org/wiki/Wikidata:SPARQL_query_service/WDQS_backend_update/October_2023_scaling_update][https://www.mediawiki.org/wiki/Wikidata_Query_Service/User_Manual#Federation]

'''Problems'''
*There was a bug that affected the appearance of visited links when using mobile device to access wiki sites. It made the links appear black; [[phab:T356928|this issue]] is fixed.  

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-14|en}}. It will be on all wikis from {{#time:j xg|2024-02-15|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] As work continues on the grid engine deprecation,[https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation] tools on the grid engine will be stopped starting on February 14th, 2024. If you have tools actively migrating you can ask for an extension so they are not stopped. [https://wikitech.wikimedia.org/wiki/Portal:Toolforge/About_Toolforge#Communication_and_support] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 05:48, 13 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26223994 --&gt;

== Tech News: 2024-08 ==

&lt;section begin=&quot;technews-2024-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/08|Translations]] are available.

'''Recent changes'''
* If you have the &quot;{{int:Tog-enotifwatchlistpages}}&quot; option enabled, edits by bot accounts no longer trigger notification emails. Previously, only minor edits would not trigger the notification emails. [https://phabricator.wikimedia.org/T356984]
* There are changes to how user and site scripts load for [[mw:Special:MyLanguage/Skin:Vector/2022| Vector 2022]] on specific wikis. The changes impacted the following Wikis: all projects with [[mw:Special:MyLanguage/Skin:Vector|Vector legacy]] as the default skin, Wikivoyage, and Wikibooks. Other wikis will be affected over the course of the next three months. Gadgets are not impacted. If you have been affected or want to minimize the impact on your project, see [[Phab:T357580| this ticket]]. Please coordinate and take action proactively.
*Newly auto-created accounts (the accounts you get when you visit a new wiki) now have the same local notification preferences as users who freshly register on that wiki. It is effected in four notification types listed in the [[phab:T353225|task's description]].
*The maximum file size when using [[c:Special:MyLanguage/Commons:Upload_Wizard|Upload Wizard]] is now 5 GiB. [https://phabricator.wikimedia.org/T191804]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-21|en}}. It will be on all wikis from {{#time:j xg|2024-02-22|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Selected tools on the grid engine have been [[wikitech:News/Toolforge_Grid_Engine_deprecation|stopped]] as we prepare to shut down the grid on March 14th, 2024. The tool's code and data have not been deleted. If you are a maintainer and you want your tool re-enabled reach out to the [[wikitech:Portal:Toolforge/About_Toolforge#Communication_and_support|team]]. Only tools that have asked for extension are still running on the grid.
* The CSS &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[https://developer.mozilla.org/en-US/docs/Web/CSS/filter filter]&lt;/code&gt;&lt;/bdi&gt; property can now be used in HTML &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;style&lt;/code&gt;&lt;/bdi&gt; attributes in wikitext. [https://phabricator.wikimedia.org/T308160]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:36, 19 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26254282 --&gt;

== Tech News: 2024-09 ==

&lt;section begin=&quot;technews-2024-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/09|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/VisualEditor_on_mobile|mobile visual editor]] is now the default editor for users who never edited before, at a small group of wikis. [[mw:Special:MyLanguage/VisualEditor_on_mobile/VE_mobile_default#A/B_test_results| Research ]] shows that users using this editor are slightly more successful publishing the edits they started, and slightly less successful publishing non-reverted edits. Users who defined the wikitext editor as their default on desktop will get the wikitext editor on mobile for their first edit on mobile as well. [https://phabricator.wikimedia.org/T352127]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/ResourceLoader/Core modules#mw.config|mw.config]] value &lt;code&gt;wgGlobalGroups&lt;/code&gt; now only contains groups that are active in the wiki. Scripts no longer have to check whether the group is active on the wiki via an API request. A code example of the above is: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;if (/globalgroupname/.test(mw.config.get(&quot;wgGlobalGroups&quot;)))&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T356008]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-28|en}}. It will be on all wikis from {{#time:j xg|2024-02-29|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* The right to change [[mw:Special:MyLanguage/Manual:Tags|edit tags]] (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;changetags&lt;/code&gt;&lt;/bdi&gt;) will be removed from users in Wikimedia sites, keeping it by default for admins and bots only. Your community can ask to retain the old configuration on your wiki before this change happens. Please indicate in [[phab:T355639|this ticket]] to keep it for your community before the end of March 2024.
'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:23, 26 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26294125 --&gt;

== Tech News: 2024-10 ==

&lt;section begin=&quot;technews-2024-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/10|Translations]] are available.

'''Recent changes'''
* The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Special:Book&lt;/code&gt;&lt;/bdi&gt; page (as well as the associated &quot;Create a book&quot; functionality) provided by the old [[mw:Special:MyLanguage/Extension:Collection|Collection extension]] has been removed from all Wikisource wikis, as it was broken. This does not affect the ability to download normal books, which is provided by the [[mw:Special:MyLanguage/Extension:Wikisource|Wikisource extension]]. [https://phabricator.wikimedia.org/T358437]
* [[m:Wikitech|Wikitech]] now uses the next-generation [[mw:Special:MyLanguage/Parsoid|Parsoid]] wikitext parser by default to generate all pages in the Talk namespace.  Report any problems on the [[mw:Talk:Parsoid/Parser_Unification/Known_Issues|Known Issues discussion page]]. You can use the [[mw:Special:MyLanguage/Extension:ParserMigration|ParserMigration]] extension to control the use of Parsoid; see the [[mw:Special:MyLanguage/Help:Extension:ParserMigration|ParserMigration help documentation]] for more details.
* Maintenance on [https://etherpad.wikimedia.org etherpad] is completed. If you encounter any issues, please indicate in [[phab:T316421|this ticket]].
* [[File:Octicons-tools.svg|12px|link=|alt=| Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] allow interface admins to create custom features with CSS and JavaScript. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Gadget&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Gadget_definition&lt;/code&gt;&lt;/bdi&gt; namespaces and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;gadgets-definition-edit&lt;/code&gt;&lt;/bdi&gt; user right were reserved for an experiment in 2015, but were never used. These were visible on Special:Search and Special:ListGroupRights. The unused namespaces and user rights are now removed. No pages are moved, and no changes need to be made. [https://phabricator.wikimedia.org/T31272]
* A usability improvement to the &quot;Add a citation&quot; in Wikipedia workflow has been made, the insert button was moved to the popup header. [https://phabricator.wikimedia.org/T354847]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-06|en}}. It will be on all wikis from {{#time:j xg|2024-03-07|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
*  All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233]
* The HTML markup of headings and section edit links will be changed later this year to improve accessibility. See [[mw:Special:MyLanguage/Heading_HTML_changes|Heading HTML changes]] for details. The new markup will be the same as in the new Parsoid wikitext parser. You can test your gadget or stylesheet with the new markup if you add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;?useparsoid=1&lt;/code&gt;&lt;/bdi&gt; to your URL ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Selecting_a_parser_using_a_URL_query_string|more info]]) or turn on Parsoid read views in your user options ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Enabling_via_user_preference|more info]]).
*

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:47, 4 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26329807 --&gt;

== Tech News: 2024-11 ==

&lt;section begin=&quot;technews-2024-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/11|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-13|en}}. It will be on all wikis from {{#time:j xg|2024-03-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* After consulting with various communities, the line height of the text on the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] will be increased to its previous value of 1.65. Different options for typography can also be set using the options in the menu, as needed. [https://phabricator.wikimedia.org/T358498]
*The active link color in [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva]] will be changed to provide more consistency with our other platforms and best practices. [https://phabricator.wikimedia.org/T358516]
* [[c:Special:MyLanguage/Commons:Structured data|Structured data on Commons]] will no longer ask whether you want to leave the page without saving. This will prevent the “information you’ve entered may not be saved” popups from appearing when no information have been entered. It will also make file pages on Commons load faster in certain cases. However, the popups will be hidden even if information has indeed been entered. If you accidentally close the page before saving the structured data you entered, that data will be lost. [https://phabricator.wikimedia.org/T312315]

'''Future changes'''
* All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:04, 11 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26374013 --&gt;

== Tech News: 2024-12 ==

&lt;section begin=&quot;technews-2024-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/12|Translations]] are available.

'''Recent changes'''
* The notice &quot;Language links are at the top of the page&quot; that appears in the [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]] main menu has been removed now that users have learned the new location of the Language switcher. [https://phabricator.wikimedia.org/T353619]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/IP_Editing:_Privacy_Enhancement_and_Abuse_Mitigation/IP_Info_feature|IP info feature]] displays data from Spur, an IP addresses database. Previously, the only data source for this feature was MaxMind. Now, IP info is more useful for patrollers. [https://phabricator.wikimedia.org/T341395]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge Grid Engine services have been shut down after the final migration process from Grid Engine to Kubernetes. [https://wikitech.wikimedia.org/wiki/Obsolete:Toolforge/Grid][https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation][https://techblog.wikimedia.org/2022/03/14/toolforge-and-grid-engine/]
* Communities can now customize the default reasons for undeleting a page by creating [[MediaWiki:Undelete-comment-dropdown]]. [https://phabricator.wikimedia.org/T326746]

'''Problems'''
* [[m:Special:MyLanguage/WMDE_Technical_Wishes/RevisionSlider|RevisionSlider]] is an interface to interactively browse a page's history. Users in [[mw:Special:MyLanguage/Extension:RevisionSlider/Developing_a_RTL-accessible_feature_in_MediaWiki_-_what_we%27ve_learned_while_creating_the_RevisionSlider|right-to-left]] languages reported RevisionSlider reacting wrong to mouse clicks. This should be fixed now. [https://phabricator.wikimedia.org/T352169]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-20|en}}. It will be on all wikis from {{#time:j xg|2024-03-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* All wikis will be read-only for a few minutes on March 20. This is planned at [https://zonestamp.toolforge.org/1710943200 14:00 UTC]. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:39, 18 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26410165 --&gt;

== Tech News: 2024-13 ==

&lt;section begin=&quot;technews-2024-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/13|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] An update was made on March 18th 2024 to how various projects load site, user JavaScript and CSS in [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]]. A [[phab:T360384|checklist]] is provided for site admins to follow.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-27|en}}. It will be on all wikis from {{#time:j xg|2024-03-28|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:56, 25 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26446209 --&gt;

== Tech News: 2024-14 ==

&lt;section begin=&quot;technews-2024-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/14|Translations]] are available.

'''Recent changes'''
* Users of the [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading|reading accessibility]] beta feature will notice that the default line height for the standard and large text options has changed. [https://phabricator.wikimedia.org/T359030]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-03|en}}. It will be on all wikis from {{#time:j xg|2024-04-04|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* The Wikimedia Foundation has an annual plan. The annual plan decides what the Wikimedia Foundation will work on. You can now read [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product &amp; Technology OKRs#Draft Key Results|the draft key results]] for the Product and Technology department. They are suggestions for what results the Foundation wants from big technical changes from July 2024 to June 2025. You can [[m:Talk:Wikimedia Foundation Annual Plan/2024-2025/Product &amp; Technology OKRs|comment on the talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 03:36, 2 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26462933 --&gt;

== Tech News: 2024-15 ==

&lt;section begin=&quot;technews-2024-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/15|Translations]] are available.

'''Recent changes'''
* Web browsers can use tools called [[:w:en:Browser extension|extensions]]. There is now a Chrome extension called [[m:Future Audiences/Experiment:Citation Needed|Citation Needed]] which you can use to see if an online statement is supported by a Wikipedia article. This is a small experiment to see if Wikipedia can be used this way. Because it is a small experiment, it can only be used in Chrome in English.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A new [[mw:Special:MyLanguage/Help:Edit Recovery|Edit Recovery]] feature has been added to all wikis, available as a [[Special:Preferences#mw-prefsection-editing|user preference]]. Once you enable it, your in-progress edits will be stored in your web browser, and if you accidentally close an editing window or your browser or computer crashes, you will be prompted to recover the unpublished text. Please leave any feedback on the [[m:Special:MyLanguage/Talk:Community Wishlist Survey 2023/Edit-recovery feature|project talk page]]. This was the #8 wish in the 2023 Community Wishlist Survey.
* Initial results of [[mw:Special:MyLanguage/Edit check|Edit check]] experiments [[mw:Special:MyLanguage/Edit_check#4_April_2024|have been published]]. Edit Check is now deployed as a default feature at [[phab:T342930#9538364|the wikis that tested it]]. [[mw:Talk:Edit check|Let us know]] if you want your wiki to be part of the next deployment of Edit check. [https://phabricator.wikimedia.org/T342930][https://phabricator.wikimedia.org/T361727]
* Readers using the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] on mobile will notice there has been an improvement in the line height across all typography settings. [https://phabricator.wikimedia.org/T359029]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-10|en}}. It will be on all wikis from {{#time:j xg|2024-04-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* New accounts and logged-out users will get the [[mw:Special:MyLanguage/VisualEditor|visual editor]] as their default editor on mobile. This deployment is made at all wikis except for the English Wikipedia. [https://phabricator.wikimedia.org/T361134]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:37, 8 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26564838 --&gt;

== Tech News: 2024-16 ==

&lt;section begin=&quot;technews-2024-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/16|Translations]] are available.

'''Problems'''
* Between 2 April and 8 April, on wikis using [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged Revisions]], the &quot;{{Int:tag-mw-reverted}}&quot; tag was not applied to undone edits. In addition, page moves, protections and imports were not autoreviewed. This problem is now fixed. [https://phabricator.wikimedia.org/T361918][https://phabricator.wikimedia.org/T361940]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-17|en}}. It will be on all wikis from {{#time:j xg|2024-04-18|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[mw:Special:MyLanguage/Help:Magic words#DEFAULTSORT|Default category sort keys]] will now affect categories added by templates placed in [[mw:Special:MyLanguage/Help:Cite|footnotes]]. Previously footnotes used the page title as the default sort key even if a different default sort key was specified (category-specific sort keys already worked). [https://phabricator.wikimedia.org/T40435]
* A new variable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;page_last_edit_age&lt;/code&gt;&lt;/bdi&gt; will be added to [[Special:AbuseFilter|abuse filters]]. It tells how many seconds ago the last edit to a page was made. [https://phabricator.wikimedia.org/T269769]

'''Future changes'''
* Volunteer developers are kindly asked to update the code of their tools and features to handle [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]]. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Four database fields will be removed from database replicas (including [[quarry:|Quarry]]). This affects only the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;abuse_filter&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;abuse_filter_history&lt;/code&gt;&lt;/bdi&gt; tables. Some queries might need to be updated. [https://phabricator.wikimedia.org/T361996]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:29, 15 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26564838 --&gt;

== Tech News: 2024-17 ==

&lt;section begin=&quot;technews-2024-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/17|Translations]] are available.

'''Recent changes'''
* Starting this week, newcomers editing Wikipedia [[mw:Special:MyLanguage/Growth/Positive reinforcement#Leveling up 3|will be encouraged]] to try structured tasks. [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer tasks|Structured tasks]] have been shown to [[mw:Special:MyLanguage/Growth/Personalized first day/Structured tasks/Add a link/Experiment analysis, December 2021|improve newcomer activation and retention]]. [https://phabricator.wikimedia.org/T348086]
* You can [[m:Special:MyLanguage/Coolest Tool Award|nominate your favorite tools]] for the fifth edition of the Coolest Tool Award. Nominations will be open until May 10. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-24|en}}. It will be on all wikis from {{#time:j xg|2024-04-25|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* This is the last warning that by the end of May 2024 the Vector 2022 skin will no longer share site and user scripts/styles with old Vector. For user-scripts that you want to keep using on Vector 2022, copy the contents of [[{{#special:MyPage}}/vector.js]] to [[{{#special:MyPage}}/vector-2022.js]]. There are [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|more technical details]] available. Interface administrators who foresee this leading to lots of technical support questions may wish to send a mass message to your community, as was done on French Wikipedia. [https://phabricator.wikimedia.org/T362701]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:28, 22 April 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26647188 --&gt;

== Tech News: 2024-18 ==

&lt;section begin=&quot;technews-2024-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/18|Translations]] are available.

'''Recent changes'''

[[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* The appearance of talk pages changed for the following wikis: {{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-viwiki/en}}. These wikis participated to a test, where 50% of users got the new design, for one year. As this test [[Mw:Special:MyLanguage/Talk pages project/Usability/Analysis|gave positive results]], the new design is deployed on these wikis as the default design. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] (&quot;{{int:discussiontools-preference-visualenhancements}}&quot;). The deployment will happen at all wikis in the coming weeks. [https://phabricator.wikimedia.org/T341491]
* Seven new wikis have been created: 
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q33014|Betawi]] ([[w:bew:|&lt;code&gt;w:bew:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T357866]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35708|Kusaal]] ([[w:kus:|&lt;code&gt;w:kus:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T359757]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35513|Igala]] ([[w:igl:|&lt;code&gt;w:igl:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T361644]
** a {{int:project-localized-name-group-wiktionary}} in [[d:Q33541|Karakalpak]] ([[wikt:kaa:|&lt;code&gt;wikt:kaa:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T362135]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q9228|Burmese]] ([[s:my:|&lt;code&gt;s:my:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T361085]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q9237|Malay]] ([[s:ms:|&lt;code&gt;s:ms:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T363039]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q8108|Georgian]] ([[s:ka:|&lt;code&gt;s:ka:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T363085]
* You can now [https://translatewiki.net/wiki/Support#Early_access:_Watch_Message_Groups_on_Translatewiki.net watch message groups/projects] on [[m:Special:MyLanguage/translatewiki.net|Translatewiki.net]]. Initially, this feature will notify you of added or deleted messages in these groups. [https://phabricator.wikimedia.org/T348501]
* Dark mode is now available on all wikis, on mobile web for logged-in users who opt into the [[Special:MobileOptions|advanced mode]]. This is the early release of the feature. Technical editors are invited to [https://night-mode-checker.wmcloud.org/ check for accessibility issues on wikis]. See [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-04|more detailed guidelines]].

'''Problems'''
* [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps can use an alternative visual style without labels, by using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mapstyle=&quot;osm&quot;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This wasn't working in previews, creating the wrong impression that it wasn't supported. This has now been fixed. [https://phabricator.wikimedia.org/T362531]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-01|en}}. It will be on all wikis from {{#time:j xg|2024-05-02|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 03:33, 30 April 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26689057 --&gt;

== Tech News: 2024-19 ==

&lt;section begin=&quot;technews-2024-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/19|Translations]] are available.

'''Recent changes'''
[[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* The appearance of talk pages changed for all wikis, except for Commons, Wikidata and most Wikipedias ([[m:Special:MyLanguage/Tech/News/2024/18|a few]] have already received this design change). You can read the detail of the changes [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] (&quot;{{int:discussiontools-preference-visualenhancements}}&quot;). The deployment will happen at remaining wikis in the coming weeks. [https://phabricator.wikimedia.org/T352087][https://phabricator.wikimedia.org/T319146]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Interface admins now have greater control over the styling of article components on mobile with the introduction of the &lt;code&gt;SiteAdminHelper&lt;/code&gt;. More information on how styles can be disabled can be found [[mw:Special:MyLanguage/Extension:WikimediaMessages#Site_admin_helper|at the extension's page]]. [https://phabricator.wikimedia.org/T363932]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] has added article body sections in JSON format and a curated short description field to the existing parsed Infobox. This expansion to the API is also available via Wikimedia Cloud Services. [https://enterprise.wikimedia.com/blog/article-sections-and-description/]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-08|en}}. It will be on all wikis from {{#time:j xg|2024-05-09|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* When you look at the Special:Log page, the first view is labelled &quot;All public logs&quot;, but it only shows some logs. This label will now say &quot;Main public logs&quot;. [https://phabricator.wikimedia.org/T237729]

'''Future changes'''
* A new service will be built to replace [[mw:Special:MyLanguage/Extension:Graph|Extension:Graph]]. Details can be found in [[mw:Special:MyLanguage/Extension:Graph/Plans|the latest update]] regarding this extension.
* Starting May 21, English Wikipedia and German Wikipedia will get the possibility to activate &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot;. This is part of the [[phab:T304110|progressive deployment of this tool to all Wikipedias]]. These communities can [[mw:Special:MyLanguage/Growth/Community configuration|activate and configure the feature locally]]. [https://phabricator.wikimedia.org/T308144]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:44, 6 May 2024 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26729363 --&gt;

== Tech News: 2024-20 ==

&lt;section begin=&quot;technews-2024-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/20|Translations]] are available.

'''Recent changes'''
* On Wikisource there is a special page listing pages of works without corresponding scan images. Now you can use the new magic word &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;__EXPECTWITHOUTSCANS__&lt;/code&gt;&lt;/bdi&gt; to exclude certain pages (list of editions or translations of works) from that list. [https://phabricator.wikimedia.org/T344214]
* If you use the [[Special:Preferences#mw-prefsection-editing|user-preference]] &quot;{{int:tog-uselivepreview}}&quot;, then the template-page feature &quot;{{int:Templatesandbox-editform-legend}}&quot; will now also work without reloading the page. [https://phabricator.wikimedia.org/T136907]
* [[mw:Special:Mylanguage/Extension:Kartographer|Kartographer]] maps can now specify an alternative text via the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;alt=&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute. This is identical in usage to the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;alt=&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute in the [[mw:Special:MyLanguage/Help:Images#Syntax|image and gallery syntax]]. An exception for this feature is wikis like Wikivoyage where the miniature maps are interactive. [https://phabricator.wikimedia.org/T328137]
* The old [[mw:Special:MyLanguage/Extension:GuidedTour|Guided Tour]] for the &quot;[[mw:Special:MyLanguage/Edit Review Improvements/New filters for edit review|New Filters for Edit Review]]&quot; feature has been removed. It was created in 2017 to show people with older accounts how the interface had changed, and has now been seen by most of the intended people. [https://phabricator.wikimedia.org/T217451]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-15|en}}. It will be on all wikis from {{#time:j xg|2024-05-16|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[{{#special:search}}]] results page will now use CSS flex attributes, for better accessibility, instead of a table. If you have a gadget or script that adjusts search results, you should update your script to the new HTML structure. [https://phabricator.wikimedia.org/T320295]

'''Future changes'''
* In the Vector 2022 skin, main pages will be displayed at full width (like special pages). The goal is to keep the number of characters per line large enough. This is related to the coming changes to typography in Vector 2022. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates|Learn more]]. [https://phabricator.wikimedia.org/T357706]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Two columns of the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Manual:pagelinks table|pagelinks]]&lt;/code&gt;&lt;/bdi&gt; database table (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_namespace&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_title&lt;/code&gt;&lt;/bdi&gt;) are being dropped soon. Users must use two columns of the new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:special:MyLanguage/Manual:linktarget table|linktarget]]&lt;/code&gt;&lt;/bdi&gt; table instead (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_namespace&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_title&lt;/code&gt;&lt;/bdi&gt;). In your existing SQL queries:
*# Replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN pagelinks&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN linktarget&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_&lt;/code&gt;&lt;/bdi&gt; in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;ON&lt;/code&gt;&lt;/bdi&gt; statement
*# Below that add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN pagelinks ON lt_id = pl_target_id&lt;/code&gt;&lt;/bdi&gt;
** See &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[phab:T222224]]&lt;/bdi&gt; for technical reasoning. [https://phabricator.wikimedia.org/T222224][https://phabricator.wikimedia.org/T299947]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:58, 13 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26762074 --&gt;

== Tech News: 2024-21 ==

&lt;section begin=&quot;technews-2024-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/21|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, will now correctly delete pages which were moved to another title. [https://phabricator.wikimedia.org/T43351]
* New changes have been made to the UploadWizard in Wikimedia Commons: the overall layout has been improved, by following new styling and spacing for the form and its fields; the headers and helper text for each of the fields was changed; the Caption field is now a required field, and there is an option for users to copy their caption into the media description. [https://commons.wikimedia.org/wiki/Commons:WMF_support_for_Commons/Upload_Wizard_Improvements#Changes_to_%22Describe%22_workflow][https://phabricator.wikimedia.org/T361049]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-22|en}}. It will be on all wikis from {{#time:j xg|2024-05-23|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]]  The HTML used to render all headings [[mw:Heading_HTML_changes|is being changed to improve accessibility]]. It will change on 22 May in some skins (Timeless, Modern, CologneBlue, Nostalgia, and Monobook). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in all other skins. The developers are also considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W21&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:04, 20 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26786311 --&gt;

== Tech News: 2024-22 ==

&lt;section begin=&quot;technews-2024-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/22|Translations]] are available.

'''Recent changes'''
* Several bugs related to the latest updates to the UploadWizard on Wikimedia Commons have been fixed. For more information, see [[:phab:T365107|T365107]] and [[:phab:T365119|T365119]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In March 2024 a new [[mw:ResourceLoader/Core_modules#addPortlet|addPortlet]] API was added to allow gadgets to create new portlets (menus) in the skin. In certain skins this can be used to create dropdowns. Gadget developers are invited to try it and [[phab:T361661|give feedback]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Some CSS in the Minerva skin has been removed to enable easier community configuration. Interface editors should check the rendering on mobile devices for aspects related to the classes: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.collapsible&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.multicol&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.reflist&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.coordinates&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.topicon&lt;/code&gt;&lt;/bdi&gt;. [[phab:T361659|Further details are available on replacement CSS]] if it is needed.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-29|en}}. It will be on all wikis from {{#time:j xg|2024-05-30|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* When you visit a wiki where you don't yet have a local account, local rules such as edit filters can sometimes prevent your account from being created. Starting this week, MediaWiki takes your global rights into account when evaluating whether you can override such local rules. [https://phabricator.wikimedia.org/T316303]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:15, 28 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26832205 --&gt;

== Tech News: 2024-23 ==

&lt;section begin=&quot;technews-2024-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/23|Translations]] are available.

'''Recent changes'''
* It is now possible for local administrators to add new links to the bottom of the site Tools menu without JavaScript. [[mw:Manual:Interface/Sidebar#Add or remove toolbox sections|Documentation is available]]. [https://phabricator.wikimedia.org/T6086]
* The message name for the definition of the tracking category of WikiHiero has changed from &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MediaWiki:Wikhiero-usage-tracking-category&lt;/code&gt;&lt;/bdi&gt;&quot; to &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MediaWiki:Wikihiero-usage-tracking-category&lt;/code&gt;&lt;/bdi&gt;&quot;. [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/wikihiero/+/1035855]
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q5317225|Kadazandusun]] ([[w:dtp:|&lt;code&gt;w:dtp:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T365220]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-05|en}}. It will be on all wikis from {{#time:j xg|2024-06-06|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* Next week, on wikis with the Vector 2022 skin as the default, logged-out desktop users will be able to choose between different font sizes. The default font size will also be increased for them. This is to make Wikimedia projects easier to read. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-06 deployments|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:35, 3 June 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26844397 --&gt;

== Tech News: 2024-24 ==

&lt;section begin=&quot;technews-2024-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/24|Translations]] are available.

'''Recent changes'''
* The software used to render SVG files has been updated to a new version, fixing many longstanding bugs in SVG rendering. [https://phabricator.wikimedia.org/T265549]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML used to render all headings [[mw:Heading HTML changes|is being changed to improve accessibility]]. It was changed last week in some skins (Vector legacy and Minerva). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in Vector-2022. The developers are still considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML markup used for citations by [[mw:Special:MyLanguage/Parsoid|Parsoid]] changed last week. In places where Parsoid previously added the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw-reference-text&lt;/code&gt;&lt;/bdi&gt; class, Parsoid now also adds the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;reference-text&lt;/code&gt;&lt;/bdi&gt; class for better compatibility with the legacy parser. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1036705]

'''Problems'''
* There was a bug with the Content Translation interface that caused the tools menus to appear in the wrong location. This has now been fixed. [https://phabricator.wikimedia.org/T366374]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-12|en}}. It will be on all wikis from {{#time:j xg|2024-06-13|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The new version of MediaWiki includes another change to the HTML markup used for citations: [[mw:Special:MyLanguage/Parsoid|Parsoid]] will now generate a &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;span class=&quot;mw-cite-backlink&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; wrapper for both named and unnamed references for better compatibility with the legacy parser. Interface administrators should verify that gadgets that interact with citations are compatible with the new markup. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1035809] 
* On multilingual wikis that use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;translate&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; system, there is a feature that shows potentially-outdated translations with a pink background until they are updated or confirmed. From this week, confirming translations will be logged, and there is a new user-right that can be required for confirming translations if the community [[m:Special:MyLanguage/Requesting wiki configuration changes|requests it]]. [https://phabricator.wikimedia.org/T49177]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:20, 10 June 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26893898 --&gt;

== Tech News: 2024-25 ==

&lt;section begin=&quot;technews-2024-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/25|Translations]] are available.

'''Recent changes'''
* People who attempt to add an external link in the visual editor will now receive immediate feedback if they attempt to link to a domain that a project has decided to block. Please see [[mw:Special:MyLanguage/Edit_check#11_June_2024|Edit check]] for more details. [https://phabricator.wikimedia.org/T366751]
* The new [[mw:Special:MyLanguage/Extension:CommunityConfiguration|Community Configuration extension]] is available [[testwiki:Special:CommunityConfiguration|on Test Wikipedia]]. This extension allows communities to customize specific features to meet their local needs. Currently only Growth features are configurable, but the extension will support other [[mw:Special:MyLanguage/Community_configuration#Use_cases|Community Configuration use cases]] in the future. [https://phabricator.wikimedia.org/T323811][https://phabricator.wikimedia.org/T360954]
* The dark mode [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] is now available on category and help pages, as well as more special pages. There may be contrast issues. Please report bugs on the [[mw:Talk:Reading/Web/Accessibility_for_reading|project talk page]]. [https://phabricator.wikimedia.org/T366370]

'''Problems'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Cloud Services tools were not available for 25 minutes last week. This was caused by a faulty hardware cable in the data center. [https://wikitech.wikimedia.org/wiki/Incidents/2024-06-11_WMCS_Ceph]
* Last week, styling updates were made to the Vector 2022 skin. This caused unforeseen issues with templates, hatnotes, and images. Changes to templates and hatnotes were reverted. Most issues with images were fixed. If you still see any, [[phab:T367463|report them here]]. [https://phabricator.wikimedia.org/T367480]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-19|en}}. It will be on all wikis from {{#time:j xg|2024-06-20|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting June 18, the [[mw:Special:MyLanguage/Help:Edit check#ref|Reference Edit Check]] will be deployed to [[phab:T361843|a new set of Wikipedias]]. This feature is intended to help newcomers and to assist edit-patrollers by inviting people who are adding new content to a Wikipedia article to add a citation when they do not do so themselves. During [[mw:Special:MyLanguage/Edit_check#Reference_Check_A/B_Test|a test at 11 wikis]], the number of citations added [https://diff.wikimedia.org/?p=127553 more than doubled] when Reference Check was shown to people. Reference Check is [[mw:Special:MyLanguage/Edit check/Configuration|community configurable]]. [https://phabricator.wikimedia.org/T361843]&lt;!-- NOTE: THE DIFF BLOG WILL BE PUBLISHED ON MONDAY --&gt;
* [[m:Special:MyLanguage/Mailing_lists|Mailing lists]] will be unavailable for roughly two hours on Tuesday 10:00–12:00 UTC. This is to enable migration to a new server and upgrade its software. [https://phabricator.wikimedia.org/T367521]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:48, 17 June 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26911987 --&gt;

== Tech News: 2024-26 ==

&lt;section begin=&quot;technews-2024-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/26|Translations]] are available.

'''Recent changes'''
* Editors will notice that there have been some changes to the background color of text in the diff view, and the color of the byte-change numbers, last week. These changes are intended to make text more readable in both light mode and dark mode, and are part of a larger effort to increase accessibility. You can share your comments or questions [[mw:Talk:Reading/Web/Accessibility for reading|on the project talkpage]]. [https://phabricator.wikimedia.org/T361717]
* The text colors that are used for visited-links, hovered-links, and active-links, were also slightly changed last week to improve their accessibility in both light mode and dark mode. [https://phabricator.wikimedia.org/T366515]

'''Problems'''
* You can [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|copy permanent links to talk page comments]] by clicking on a comment's timestamp. [[mw:Talk pages project/Permalinks|This feature]] did not always work when the topic title was very long and the link was used as a wikitext link. This has been fixed. Thanks to Lofhi for submitting the bug. [https://phabricator.wikimedia.org/T356196]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-26|en}}. It will be on all wikis from {{#time:j xg|2024-06-27|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting 26 June, all talk pages messages' timestamps will become a link at English Wikipedia, making this feature available for you to use at all wikis. This link is a permanent link to the comment. It allows users to find the comment they were linked to, even if this comment has since been moved elsewhere. You can read more about this feature [[DiffBlog:/2024/01/29/talk-page-permalinks-dont-lose-your-threads/|on Diff]] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|on Mediawiki.org]]. [https://phabricator.wikimedia.org/T365974]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:32, 24 June 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26989424 --&gt;

== Tech News: 2024-27 ==

&lt;section begin=&quot;technews-2024-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/27|Translations]] are available.

'''Recent changes'''
* Over the next three weeks, dark mode will become available for all users, both logged-in and logged-out, starting with the mobile web version. This fulfils one of the [[m:Special:MyLanguage/Community_Wishlist_Survey_2023/Reading/Dark_mode|top-requested community wishes]], and improves low-contrast reading and usage in low-light settings. As part of these changes, dark mode will also work on User-pages and Portals. There is more information in [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading/Updates#June_2024:_Typography_and_dark_mode_deployments,_new_global_preferences|the latest Web team update]]. [https://phabricator.wikimedia.org/T366364]
* Logged-in users can now set [[m:Special:GlobalPreferences#mw-prefsection-rendering-skin-skin-prefs|global preferences for the text-size and dark-mode]], thanks to a combined effort across Foundation teams. This allows Wikimedians using multiple wikis to set up a consistent reading experience easily, for example by switching between light and dark mode only once for all wikis. [https://phabricator.wikimedia.org/T341278] 
* If you use a very old web browser some features might not work on the Wikimedia wikis. This affects Internet Explorer 11 and versions of Chrome, Firefox and Safari older than 2016. This change makes it possible to use new [[d:Q46441|CSS]] features and to send less code to all readers. [https://phabricator.wikimedia.org/T288287][https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:How_to_make_a_MediaWiki_skin#Using_CSS_variables_for_supporting_different_themes_e.g._dark_mode]
* Wikipedia Admins can customize local wiki configuration options easily using [[mw:Special:MyLanguage/Community Configuration|Community Configuration]]. Community Configuration was created to allow communities to customize how some features work, because each language wiki has unique needs. At the moment, admins can configure [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] on their home wikis, in order to better recruit and retain new editors. More options will be provided in the coming months. [https://phabricator.wikimedia.org/T366458]
* Editors interested in language issues that are related to [[w:en:Unicode|Unicode standards]], can now discuss those topics at [[mw:Talk:WMF membership with Unicode Consortium|a new conversation space in MediaWiki.org]]. The Wikimedia Foundation is now a [[mw:Special:MyLanguage/WMF membership with Unicode Consortium|member of the Unicode Consortium]], and the coordination group can collaboratively review the issues discussed and, where appropriate, bring them to the attention of the Unicode Consortium.
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2891049|Mandailing]] ([[w:btm:|&lt;code&gt;w:btm:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T368038] 

'''Problems'''
* Editors can once again click on links within the visual editor's citation-preview, thanks to a bug fix by the Editing Team. [https://phabricator.wikimedia.org/T368119]

'''Future changes'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 2 weeks. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:59, 1 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27038456 --&gt;

== Tech News: 2024-28 ==

&lt;section begin=&quot;technews-2024-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/28|Translations]] are available.

'''Recent changes'''
* At the Wikimedia Foundation a new task force was formed to replace the disabled Graph with [[mw:Special:MyLanguage/Extension:Chart/Project|more secure, easy to use, and extensible Chart]]. You can [[mw:Special:MyLanguage/Newsletter:Chart Project|subscribe to the newsletter]] to get notified about new project updates and other news about Chart.
* The [[m:Special:MyLanguage/CampaignEvents|CampaignEvents]] extension is now available on Meta-wiki, Igbo Wikipedia, and Swahili Wikipedia, and can be requested on your wiki. This extension helps in managing and making events more visible, giving Event organizers the ability to use tools like the Event registration tool. To learn more about the deployment status and how to request this extension for your wiki, visit the [[m:Special:MyLanguage/CampaignEvents/Deployment_status|CampaignEvents page on Meta-wiki]]. 
* Editors using the iOS Wikipedia app who have more than 50 edits can now use the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits#Add an image|Add an Image]] feature. This feature presents opportunities for small but useful contributions to Wikipedia.
* Thank you to [[mw:MediaWiki Product Insights/Contributor retention and growth/Celebration|all of the authors]] who have contributed to MediaWiki Core. As a result of these contributions, the [[mw:MediaWiki Product Insights/Contributor retention and growth|percentage of authors contributing more than 5 patches has increased by 25% since last year]], which helps ensure the sustainability of the platform for the Wikimedia projects.

'''Problems'''
* A problem with the color of the talkpage tabs always showing as blue, even for non-existent pages which should have been red, affecting the Vector 2022 skin, [[phab:T367982|has been fixed]]. 

'''Future changes'''
* The Trust and Safety Product team wants to introduce [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] with as little disruption to tools and workflows as possible. Volunteer developers, including gadget and user-script maintainers, are kindly asked to update the code of their tools and features to handle temporary accounts. The team has [[mw:Trust and Safety Product/Temporary Accounts/For developers|created documentation]] explaining how to do the update. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]].

'''Tech News survey'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 1 more week. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:31, 8 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27080357 --&gt;

== Tech News: 2024-29 ==

&lt;section begin=&quot;technews-2024-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/29|Translations]] are available.

'''Tech News survey'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 3 more days. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia developers can now officially continue to use both [[mw:Special:MyLanguage/Gerrit|Gerrit]] and [[mw:Special:MyLanguage/GitLab|GitLab]], due to a June 24 decision by the Wikimedia Foundation to support software development on both platforms. Gerrit and GitLab are both code repositories used by developers to write, review, and deploy the software code that supports the MediaWiki software that the wiki projects are built on, as well as the tools used by editors to create and improve content. This decision will safeguard the productivity of our developers and prevent problems in code review from affecting our users. More details are available in the [[mw:GitLab/Migration status|Migration status]] page.
* The Wikimedia Foundation seeks applicants for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product and Technology Advisory Council]] (PTAC). This group will bring technical contributors and Wikimedia Foundation together to co-define a more resilient, future-proof technological platform. Council members will evaluate and consult on the movement's product and technical activities, so that we develop multi-generational projects. We are looking for a range of technical contributors across the globe, from a variety of Wikimedia projects. [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal#Joining the PTAC as a technical volunteer|Please apply here by August 10]].
* Editors with rollback user-rights who use the Wikipedia App for Android can use the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Anti Vandalism|Edit Patrol]] features. These features include a new feed of Recent Changes, related links such as Undo and Rollback, and the ability to create and save a personal library of user talk messages to use while patrolling. If your wiki wants to make these features available to users who do not have rollback rights but have reached a certain edit threshold, [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android#Contact us|you can contact the team]]. You can [[diffblog:2024/07/10/ِaddressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/|read more about this project on Diff blog]].
* Editors who have access to [[m:Special:MyLanguage/The_Wikipedia_Library|The Wikipedia Library]] can once again use non-open access content in SpringerLinks, after the Foundation [[phab:T368865|contacted]] them to restore access. You can read more about [[m:Tech/News/Recently_resolved_community_tasks|this and 21 other community-submitted tasks that were completed last week]]. 

'''Changes later this week'''
* This week, [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-07 deployments|dark mode will be available on a number of Wikipedias]], both desktop and mobile, for logged-in and logged-out users. Interface admins and user script maintainers are encouraged to check gadgets and user scripts in the dark mode, to find any hard-coded colors and fix them. There are some [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations for dark mode compatibility]] to help.

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Next week, functionaries, volunteers maintaining tools, and software development teams are invited to test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on testwiki. Temporary accounts is a feature that will help improve privacy on the wikis. No further temporary account deployments are scheduled yet. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]]. [https://phabricator.wikimedia.org/T348895]
* Editors who upload files cross-wiki, or teach other people how to do so, may wish to join a Wikimedia Commons discussion. The Commons community is discussing limiting who can upload files through the cross-wiki upload/Upload dialog feature to users auto-confirmed on Wikimedia Commons. This is due to the large amount of copyright violations uploaded this way. There is a short summary at [[c:Special:MyLanguage/Commons:Cross-wiki upload|Commons:Cross-wiki upload]] and [[c:Commons:Village pump/Proposals#Deactivate cross-wiki uploads for new users|discussion at Commons:Village Pump]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]].
&lt;/div&gt;&lt;section end=&quot;technews-2024-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:31, 16 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27124561 --&gt;

== Tech News: 2024-30 ==

&lt;section begin=&quot;technews-2024-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/30|Translations]] are available.

'''Feature News'''
* Stewards can now [[:m:Special:MyLanguage/Global_blocks|globally block]] accounts. Before [[phab:T17294|the change]] only IP addresses and IP ranges could be blocked globally. Global account blocks are useful when the blocked user should not be logged out. [[:m:Special:MyLanguage/Global_locks|Global locks]] (a similar tool logging the user out of their account) are unaffected by this change. The new global account block feature is related to the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project, which is a new type of user account that replaces IP addresses of unregistered editors that are no longer made public.
* Later this week, Wikimedia site users will notice that the Interface of [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as &quot;Pending Changes&quot;) is improved and consistent with the rest of the MediaWiki interface and [[mw:Special:MyLanguage/Codex|Wikimedia's design system]]. The FlaggedRevs interface experience on mobile and [[mw:Special:MyLanguage/Skin:MinervaNeue|Minerva skin]] was inconsistent before it was fixed and ported to [[mw:Special:MyLanguage/Codex|Codex]] by the WMF Growth team and some volunteers. [https://phabricator.wikimedia.org/T191156]
*  Wikimedia site users can now submit account vanishing requests via [[m:Special:GlobalVanishRequest|GlobalVanishRequest]]. This feature is used when a contributor wishes to stop editing forever. It helps you hide your past association and edit to protect your privacy. Once processed, the account will be locked and renamed. [https://phabricator.wikimedia.org/T367329]
* Have you tried monitoring and addressing vandalism in Wikipedia using your phone? [https://diff.wikimedia.org/2024/07/10/%d9%90addressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/ A Diff blog post on Patrolling features in the Mobile App] highlights some of the new capabilities of the feature, including swiping through a feed of recent changes and a personal library of user talk messages for use when patrolling from your phone. 
* Wikimedia contributors and GLAM (galleries, libraries, archives, and museums) organisations can now learn and measure the impact Wikimedia Commons is having towards creating quality encyclopedic content using the [https://doc.wikimedia.org/generated-data-platform/aqs/analytics-api/reference/commons.html Commons Impact Metrics] analytics dashboard. The dashboard offers organizations analytics on things like monthly edits in a category, the most viewed files, and which Wikimedia articles are using Commons images. As a result of these new data dumps, GLAM organisation can more reliably measure their return on investment for programs bringing content into the digital Commons. [https://diff.wikimedia.org/2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/]

'''Project Updates'''
*  Come share your ideas for improving the wikis on the newly reopened [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]]. The Community Wishlist is Wikimedia’s forum for volunteers to share ideas (called wishes) to improve how the wikis work. The new version of the wishlist is always open, works with both wikitext and Visual Editor, and allows wishes in any language.

'''Learn more'''
* Have you ever wondered how Wikimedia software works across over 300 languages? This is 253 languages more than the Google Chrome interface, and it's no accident. The Language and Product Localization Team at the Wikimedia Foundation supports your work by adapting all the tools and interfaces in the MediaWiki software so that contributors in our movement who translate pages and strings can translate them and have the sites in all languages. Read more about the team and their upcoming work on [https://diff.wikimedia.org/2024/07/17/building-towards-a-robust-multilingual-knowledge-ecosystem-for-the-wikimedia-movement/ Diff]. 
* How can Wikimedia build innovative and experimental products while maintaining such heavily used websites? A recent [https://diff.wikimedia.org/2024/07/09/on-the-value-of-experimentation/ blog post] by WMF staff Johan Jönsson highlights the work of the [[m:Future Audiences#Objectives and Key Results|WMF Future Audience initiative]], where the goal is not to build polished products but test out new ideas, such as a [[m:Future_Audiences/Experiments: conversational/generative AI|ChatGPT plugin]] and [[m:Future_Audiences/Experiment:Add a Fact|Add a Fact]], to help take Wikimedia into the future. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]].
&lt;/div&gt;&lt;section end=&quot;technews-2024-W30&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:04, 23 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27142915 --&gt;

== Tech News: 2024-31 ==

&lt;section begin=&quot;technews-2024-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/31|Translations]] are available.

'''Feature news'''
* Editors using the Visual Editor in languages that use non-Latin characters for numbers, such as Hindi, Manipuri and Eastern Arabic, may notice some changes in the formatting of reference numbers. This is a side effect of preparing a new sub-referencing feature, and will also allow fixing some general numbering issues in Visual Editor. If you notice any related problems on your wiki, please share details at the [[m:Talk:WMDE Technical Wishes/Sub-referencing|project talkpage]].

'''Bugs status'''
* Some logged-in editors were briefly unable to edit or load pages last week. [[phab:T370304|These errors]] were mainly due to the addition of new [[mw:Special:MyLanguage/Help:Extension:Linter|linter]] rules which led to caching problems. Fixes have been applied and investigations are continuing.
* Editors can use the [[mw:Special:MyLanguage/Trust and Safety Product/IP Info|IP Information tool]] to get information about IP addresses. This tool is available as a Beta Feature in your preferences. The tool was not available for a few days last week, but is now working again. Thank you to Shizhao for filing the bug report. You can read about that, and [[m:Tech/News/Recently resolved community tasks#2024-07-25|28 other community-submitted tasks]] that were resolved last week.

'''Project updates'''
* There are new features and improvements to Phabricator from the Release Engineering and Collaboration Services teams, and some volunteers, including: the search systems, the new task creation system, the login systems, the translation setup which has resulted in support for more languages (thanks to Pppery), and fixes for many edge-case errors. You can [[phab:phame/post/view/316/iterative_improvements/|read details about these and other improvements in this summary]].
* There is an [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|update on the Charts project]]. The team has decided which visualization library to use, which chart types to start focusing on, and where to store chart definitions.
* One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9056|Czech]] ([[voy:cs:|&lt;code&gt;voy:cs:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T370905] 

'''Learn more'''
* There is a [[diffblog:2024/07/26/the-journey-to-open-our-first-data-center-in-south-america/|new Wikimedia Foundation data center]] in São Paulo, Brazil which helps to reduce load times.
* There is new [[diffblog:2024/07/22/the-perplexing-process-of-uploading-images-to-wikipedia/|user research]] on problems with the process of uploading images.
* Commons Impact Metrics are [[diffblog:2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/|now available]] via data dumps and API.
* The latest quarterly [[mw:Technical Community Newsletter/2024/July|Technical Community Newsletter]] is now available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:10, 29 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27164109 --&gt;

== Tech News: 2024-32 ==

&lt;section begin=&quot;technews-2024-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/32|Translations]] are available.

'''Feature news'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Two new parser functions will be available this week: &lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic_words#dir|#dir]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; and &lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic_words#bcp47|#bcp47]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt;. These will reduce the need for &lt;code&gt;Template:Dir&lt;/code&gt; and &lt;code&gt;Template:BCP47&lt;/code&gt; on Commons and allow us to [[phab:T343131|drop 100 million rows]] from the &quot;what links here&quot; database. Editors at any wiki that use these templates, can help by replacing the templates with these new functions. The templates at Commons will be updated during the Hackathon at Wikimania. [https://phabricator.wikimedia.org/T359761][https://phabricator.wikimedia.org/T366623]
* Communities can request the activation of the visual editor on entire namespaces where discussions sometimes happen (for instance ''Wikipedia:'' or ''Wikisource:'' namespaces) if they understand the [[mw:Special:MyLanguage/Help:VisualEditor/FAQ#WPNS|known limitations]]. For discussions, users can already use [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] in these namespaces.
* The tracking category &quot;Pages using Timeline&quot; has been renamed to &quot;Pages using the EasyTimeline extension&quot; [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3ATimeline-tracking-category&amp;namespace=8 in TranslateWiki]. Wikis that have created the category locally should rename their local creation to match.

'''Project updates'''
* Editors who help to organize WikiProjects and similar on-wiki collaborations, are invited to share ideas and examples of successful collaborations with the Campaigns and Programs teams. You can fill out [[m:Special:MyLanguage/Campaigns/WikiProjects|a brief survey]] or share your thoughts [[m:Talk:Campaigns/WikiProjects|on the talkpage]]. The teams are particularly looking for details about successful collaborations on non-English wikis.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The new parser is being rolled out on {{int:project-localized-name-group-wikivoyage}} wikis over the next few months. The {{int:project-localized-name-enwikivoyage}} and {{int:project-localized-name-hewikivoyage}} were [[phab:T365367|switched]] to Parsoid last week. For more information, see [[mw:Parsoid/Parser_Unification|Parsoid/Parser Unification]].

'''Learn more'''
* There will be more than 200 sessions at Wikimania this week. Here is a summary of some of the [[diffblog:2024/08/05/interested-in-product-and-tech-here-are-some-wikimania-sessions-you-dont-want-to-miss/|key sessions related to the product and technology area]].
* The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/07-02|Wikimedia Foundation Bulletin]] is available.
* The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/July|Language and Internationalization newsletter]] is available. It includes: New design previews for Translatable pages; Updates about MinT for Wiki Readers; the release of Translation dumps; and more.
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/31|Growth newsletter]] is available.
* The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/July 2024|MediaWiki Product Insights newsletter]] is available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:43, 5 August 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27233905 --&gt;

== Tech News: 2024-33 ==

&lt;section begin=&quot;technews-2024-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/33|Translations]] are available.

'''Feature news'''
* [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] editors and maintainers can now [[mw:Special:MyLanguage/Extension:AbuseFilter/Actions#Show a CAPTCHA|make a CAPTCHA show if a filter matches an edit]]. This allows communities to quickly respond to spamming by automated bots. [https://phabricator.wikimedia.org/T20110]
* [[m:Special:MyLanguage/Stewards|Stewards]] can now specify if global blocks should prevent account creation. Before [[phab:T17273|this change]] by the [[mw:Special:MyLanguage/Trust and Safety Product|Trust and Safety Product]] Team, all global blocks would prevent account creation. This will allow stewards to reduce the unintended side-effects of global blocks on IP addresses.

'''Project updates'''
* [[wikitech:Help talk:Toolforge/Toolforge standards committee#August_2024_committee_nominations|Nominations are open on Wikitech]] for new members to refresh the [[wikitech:Help:Toolforge/Toolforge standards committee|Toolforge standards committee]]. The committee oversees the Toolforge [[wikitech:Help:Toolforge/Right to fork policy|Right to fork policy]] and [[wikitech:Help:Toolforge/Abandoned tool policy|Abandoned tool policy]] among other duties. Nominations will remain open until at least 2024-08-26.
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2880037|West Coast Bajau]] ([[w:bdr:|&lt;code&gt;w:bdr:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T371757]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:21, 12 August 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27253654 --&gt;

== Tech News: 2024-34 ==

&lt;section begin=&quot;technews-2024-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/34|Translations]] are available.

'''Feature news'''
* Editors who want to re-use references but with different details such as page numbers, will be able to do so by the end of 2024, using a new [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Sub-referencing in a nutshell|sub-referencing]] feature. You can read more [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|about the project]] and [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|how to test the prototype]].
* Editors using tracking categories to identify which pages use specific extensions may notice that six of the categories have been renamed to make them more easily understood and consistent. These categories are automatically added to pages that use specialized MediaWiki extensions. The affected names are for: [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aintersection-category&amp;namespace=8 DynamicPageList], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Akartographer-tracking-category&amp;namespace=8 Kartographer], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aphonos-tracking-category&amp;namespace=8 Phonos], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Arss-tracking-category&amp;namespace=8 RSS], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Ascore-use-category&amp;namespace=8 Score], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Awikihiero-usage-tracking-category&amp;namespace=8 WikiHiero]. Wikis that have created the category locally should rename their local creation to match. Thanks to Pppery for these improvements. [https://phabricator.wikimedia.org/T347324]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Technical volunteers who edit modules and want to get a list of the categories used on a page, can now do so using the &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;categories&lt;/bdi&gt;&lt;/code&gt; property of &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[mediawikiwiki:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Title objects|mw.title objects]]&lt;/bdi&gt;&lt;/code&gt;. This enables wikis to configure workflows such as category-specific edit notices. Thanks to SD001 for these improvements. [https://phabricator.wikimedia.org/T50175][https://phabricator.wikimedia.org/T85372]

'''Bugs status'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Your help is needed to check if any pages need to be moved or deleted. A maintenance script was run to clean up unreachable pages (due to Unicode issues or introduction of new namespaces/namespace aliases). The script tried to find appropriate names for the pages (e.g. by following the Unicode changes or by moving pages whose titles on Wikipedia start with &lt;code&gt;Talk:WP:&lt;/code&gt; so that their titles start with &lt;code&gt;Wikipedia talk:&lt;/code&gt;), but it may have failed for some pages, and moved them to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[Special:PrefixIndex/T195546/]]&lt;/bdi&gt; instead. Your community should check if any pages are listed there, and move them to the correct titles, or delete them if they are no longer needed. A full log (including pages for which appropriate names could be found) is available in [[phab:P67388]].
* Editors who volunteer as [[mw:Special:MyLanguage/Help:Growth/Mentorship|mentors]] to newcomers on their wiki are once again able to access lists of potential mentees who they can connect with to offer help and guidance. This functionality was restored thanks to [[phab:T372164|a bug fix]]. Thank you to Mbch331 for filing the bug report. You can read about that, and 18 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Project updates'''
* The application deadline for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product &amp; Technology Advisory Council]] (PTAC) has been extended to September 16. Members will help by providing advice to Foundation Product and Technology leadership on short and long term plans, on complex strategic problems, and help to get feedback from more contributors and technical communities. Selected members should expect to spend roughly 5 hours per month for the Council, during the one year pilot. Please consider applying, and spread the word to volunteers you think would make a positive contribution to the committee.

'''Learn more'''
* The [[m:Special:MyLanguage/Coolest Tool Award#2024 Winners|2024 Coolest Tool Awards]] were awarded at Wikimania, in seven categories. For example, one award went to the ISA Tool, used for adding structured data to files on Commons, which was recently improved during the [[m:Event:Wiki Mentor Africa ISA Hackathon 2024|Wiki Mentor Africa Hackathon]]. You can see video demonstrations of each tool at the awards page. Congratulations to this year's recipients, and thank you to all tool creators and maintainers.
* The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/08-01|Wikimedia Foundation Bulletin]] is available, and includes some highlights from Wikimania, an upcoming Language community meeting, and other news from the movement.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W34&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:54, 20 August 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27307284 --&gt;

== Tech News: 2024-35 ==

&lt;section begin=&quot;technews-2024-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/35|Translations]] are available.

'''Feature news'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Administrators can now test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on test2wiki. This was done to allow cross-wiki testing of temporary accounts, for when temporary accounts switch between projects. The feature was enabled on testwiki a few weeks ago. No further temporary account deployments are scheduled yet. Temporary Accounts is a project to create a new type of user account that replaces IP addresses of unregistered editors which are no longer made public. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]].
* Later this week, editors at wikis that use [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as &quot;Pending Changes&quot;) may notice that the indicators at the top of articles have changed. This change makes the system more consistent with the rest of the MediaWiki interface. [https://phabricator.wikimedia.org/T191156]

'''Bugs status'''
* Editors who use the 2010 wikitext editor, and use the Character Insert buttons, will [[phab:T361465|no longer]] experience problems with the buttons adding content into the edit-summary instead of the edit-window. You can read more about that, and 26 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Project updates'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Please review and vote on [[m:Special:MyLanguage/Community Wishlist/Focus areas|Focus Areas]], which are groups of wishes that share a problem. Focus Areas were created for the newly reopened Community Wishlist, which is now open year-round for submissions. The first batch of focus areas are specific to moderator workflows, around welcoming newcomers, minimizing repetitive tasks, and prioritizing tasks. Once volunteers have reviewed and voted on focus areas, the Foundation will then review and select focus areas for prioritization.
* Do you have a project and are willing to provide a three (3) month mentorship for an intern? [[mw:Special:MyLanguage/Outreachy|Outreachy]] is a twice a year program for people to participate in a paid internship that will start in December 2024 and end in early March 2025, and they need mentors and projects to work on. Projects can be focused on coding or non-coding (design, documentation, translation, research). See the Outreachy page for more details, and a list of past projects since 2013.

'''Learn more'''
* If you're curious about the product and technology improvements made by the Wikimedia Foundation last year, read [[diffblog:2024/08/21/wikimedia-foundation-product-technology-improving-the-user-experience/|this recent highlights summary on Diff]].
* To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out:
** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Community Configuration - Shaping On-Wiki Functionality Together.webm|Community Configuration - Shaping On-Wiki Functionality Together]] (55 mins) - about the [[mw:Special:MyLanguage/Community Configuration|Community Configuration]] project.
** [[c:File:Wikimania 2024 - Belgrade - Day 1 - Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views.webm|Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views]] (30 mins) - an overview for both technical and non technical audiences, covering some of the challenges and open questions, related to the [[mw:MediaWiki Product Insights|platform evolution, stewardship and developer experiences]] research.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:33, 26 August 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27341211 --&gt;

== Tech News: 2024-36 ==

&lt;section begin=&quot;technews-2024-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/36|Translations]] are available.

'''Weekly highlight'''
* Editors and volunteer developers interested in data visualisation can now test the new software for charts. Its early version is available on beta Commons and beta Wikipedia. This is an important milestone before making charts available on regular wikis. You can [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|read more about this project update]] and help to test the charts.

'''Feature news'''
* Editors who use the [[{{#special:Unusedtemplates}}]] page can now filter out pages which are expected to be there permanently, such as sandboxes, test-cases, and templates that are always substituted. Editors can add the new magic word [[mw:Special:MyLanguage/Help:Magic words#EXPECTUNUSEDTEMPLATE|&lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;__EXPECTUNUSEDTEMPLATE__&lt;/nowiki&gt;&lt;/code&gt;]] to a template page to hide it from the listing. Thanks to Sophivorus and DannyS712 for these improvements. [https://phabricator.wikimedia.org/T184633]
* Editors who use the New Topic tool on discussion pages, will [[phab:T334163|now be reminded]] to add a section header, which should help reduce the quantity of newcomers who add sections without a header. You can read more about that, and {{formatnum:28}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].
* Last week, some Toolforge tools had occasional connection problems. The cause is still being investigated, but the problems have been resolved for now. [https://phabricator.wikimedia.org/T373243]
* Translation administrators at multilingual wikis, when editing multiple translation units, can now easily mark which changes require updates to the translation. This is possible with the [[phab:T298852#10087288|new dropdown menu]].

'''Project updates'''
* A new draft text of a policy discussing the use of Wikimedia's APIs [[m:Special:MyLanguage/API Policy Update 2024|has been published on Meta-Wiki]]. The draft text does not reflect a change in policy around the APIs; instead, it is an attempt to codify existing API rules. Comments, questions, and suggestions are welcome on [[m:Talk:API Policy Update 2024|the proposed update’s talk page]] until September 13 or until those discussions have concluded.

'''Learn more'''
* To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out:
** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Charts, the successor of Graphs - A secure and extensible tool for data visualization.webm|Charts, the successor of Graphs - A secure and extensible tool for data visualization]] (25 mins) – about the above-mentioned Charts project.
** [[c:File:Wikimania 2024 - Ohrid - Day 3 - State of Language Technology and Onboarding at Wikimedia.webm|State of Language Technology and Onboarding at Wikimedia]] (90 mins) – about some of the language tools that support Wikimedia sites, such as [[mw:Special:MyLanguage/Content translation|Content]]/[[mw:Special:MyLanguage/Content translation/Section translation|Section Translation]], [[mw:Special:MyLanguage/MinT|MinT]], and LanguageConverter; also the current state and future of languages onboarding. [https://phabricator.wikimedia.org/T368772]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:07, 3 September 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27390268 --&gt;

== Tech News: 2024-37 ==

&lt;section begin=&quot;technews-2024-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/37|Translations]] are available.

'''Feature news'''
* Starting this week, the standard [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will receive new colors that make them compatible in dark mode. This is the first of many changes to come as part of a major upgrade to syntax highlighting. You can learn more about what's to come on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]]. [https://phabricator.wikimedia.org/T365311][https://phabricator.wikimedia.org/T259059]
* Editors of wikis using Wikidata will now be notified of only relevant Wikidata changes in their watchlist. This is because the Lua functions &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;entity:getSitelink()&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw.wikibase.getSitelink(qid)&lt;/code&gt;&lt;/bdi&gt; will have their logic unified for tracking different aspects of sitelinks to reduce junk notifications from [[m:Wikidata For Wikimedia Projects/Projects/Watchlist Wikidata Sitelinks Tracking|inconsistent sitelinks tracking]]. [https://phabricator.wikimedia.org/T295356]

'''Project updates'''
*  Users of all Wikis will have access to Wikimedia sites as read-only for a few minutes on September 25, starting at 15:00 UTC. This is a planned datacenter switchover for maintenance purposes. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T370962]
* Contributors of [[phab:T363538#10123348|11 Wikipedias]], including English will have a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS&lt;/code&gt;&lt;/bdi&gt; namespace added to their Wikipedias. This improvement ensures that links beginning with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS:&lt;/code&gt;&lt;/bdi&gt; (usually shortcuts to the [[w:en:Wikipedia:Manual of Style|Manual of Style]]) are not broken by [[w:en:Mooré|Mooré]] Wikipedia (language code &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mos&lt;/code&gt;&lt;/bdi&gt;). [https://phabricator.wikimedia.org/T363538]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:52, 9 September 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27424457 --&gt;

== Tech News: 2024-38 ==

&lt;section begin=&quot;technews-2024-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/38|Translations]] are available.

'''Improvements and Maintenance'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Editors interested in templates can help by reading the latest Wishlist focus area, [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template recall and discovery]], and share your feedback on the talkpage. This input helps the Community Tech team to decide the right technical approach to build. Everyone is also encouraged to continue adding [[m:Special:MyLanguage/Community Wishlist|new wishes]].
* The new automated [[{{#special:NamespaceInfo}}]] page helps editors understand which [[mw:Special:MyLanguage/Help:Namespaces|namespaces]] exist on each wiki, and some details about how they are configured. Thanks to DannyS712 for these improvements. [https://phabricator.wikimedia.org/T263513]
* [[mw:Special:MyLanguage/Help:Edit check#Reference check|References Check]] is a feature that encourages editors to add a citation when they add a new paragraph to a Wikipedia article. For a short time, the corresponding tag &quot;Edit Check (references) activated&quot; was erroneously being applied to some edits outside of the main namespace. This has been fixed. [https://phabricator.wikimedia.org/T373692]
* It is now possible for a wiki community to change the order in which a page’s categories are displayed on their wiki. By default, categories are displayed in the order they appear in the wikitext. Now, wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool authors can now access ToolsDB's [[wikitech:Portal:Data Services#ToolsDB|public databases]] from both [[m:Special:MyLanguage/Research:Quarry|Quarry]] and [[wikitech:Superset|Superset]]. Those databases have always been accessible to every [[wikitech:Portal:Toolforge|Toolforge]] user, but they are now more broadly accessible, as Quarry can be accessed by anyone with a Wikimedia account. In addition, Quarry's internal database can now be [[m:Special:MyLanguage/Research:Quarry#Querying Quarry's own database|queried from Quarry itself]]. This database contains information about all queries that are being run and starred by users in Quarry. This information was already public through the web interface, but you can now query it using SQL. You can read more about that, and {{formatnum:20}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].
* Any pages or tools that still use the very old CSS classes &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw-message-box&lt;/code&gt;&lt;/bdi&gt; need to be updated. These old classes will be removed next week or soon afterwards. Editors can use a [https://global-search.toolforge.org/?q=mw-message-box&amp;regex=1&amp;namespaces=&amp;title= global-search] to determine what needs to be changed. It is possible to use the newer &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;cdx-message&lt;/code&gt;&lt;/bdi&gt; group of classes as a replacement (see [https://doc.wikimedia.org/codex/latest/components/demos/message.html#css-only-version the relevant Codex documentation], and [https://meta.wikimedia.org/w/index.php?title=Tech/Header&amp;diff=prev&amp;oldid=27449042 an example update]), but using locally defined onwiki classes would be best. [https://phabricator.wikimedia.org/T374499]

'''Technical project updates'''
* Next week, all Wikimedia wikis will be read-only for a few minutes. This will start on September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. This is a planned datacenter switchover for maintenance purposes. [[m:Special:MyLanguage/Tech/Server switch|This maintenance process also targets other services.]] The previous switchover took 3 minutes, and the Site Reliability Engineering teams use many tools to make sure that this essential maintenance work happens as quickly as possible. [https://phabricator.wikimedia.org/T370962]

'''Tech in depth'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/August 2024|MediaWiki Product Insights newsletter]] is available. This edition includes details about: research about [[mw:Special:MyLanguage/Manual:Hooks|hook]] handlers to help simplify development, research about performance improvements, work to improve the REST API for end-users, and more.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out: 
** [[c:File:Wikimania 2024 - Auditorium Kyiv - Day 4 - Hackathon Showcase.webm|Hackathon Showcase]] (45 mins) - 19 short presentations by some of the Hackathon participants, describing some of the projects they worked on, such as automated testing of maintenance scripts, a video-cutting command line tool, and interface improvements for various tools. There are [[phab:T369234|more details and links available]] in the Phabricator task.
** [[c:File:Co-Creating a Sustainable Future for the Toolforge Ecosystem.webm|Co-Creating a Sustainable Future for the Toolforge Ecosystem]] (40 mins) - a roundtable discussion for tool-maintainers, users, and supporters of Toolforge about how to make the platform sustainable and how to evaluate the tools available there.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:02, 17 September 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27460876 --&gt;

== Tech News: 2024-39 ==

&lt;section begin=&quot;technews-2024-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/39|Translations]] are available.

'''Weekly highlight'''
* All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only]] for a few minutes on Wednesday September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. Reading the wikis will not be interrupted, but editing will be paused. These twice-yearly processes allow WMF's site reliability engineering teams to remain prepared to keep the wikis functioning even in the event of a major interruption to one of our data centers.

'''Updates for editors'''
[[File:Add alt text from a halfsheet, with the article behind.png|thumb|A screenshot of the interface for the Alt Text suggested-edit feature]]
* Editors who use the iOS Wikipedia app in Spanish, Portuguese, French, or Chinese, may see the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|Alt Text suggested-edit experiment]] after editing an article, or completing a suggested edit using &quot;[[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project#Hypothesis 2 Add an Image Suggested Edit|Add an image]]&quot;. Alt-text helps people with visual impairments to read Wikipedia articles. The team aims to learn if adding alt-text to images is a task that editors can be successful with. Please share any feedback on [[mw:Talk:Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|the discussion page]].
* The Codex color palette has been updated with new and revised colors for the MediaWiki user interfaces. The [[mw:Special:MyLanguage/Design System Team/Color/Design documentation#Updates|most noticeable changes]] for editors include updates for: dark mode colors for Links and for quiet Buttons (progressive and destructive), visited Link colors for both light and dark modes, and background colors for system-messages in both light and dark modes.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] It is now possible to include clickable wikilinks and external links inside code blocks. This includes links that are used within &lt;code&gt;&lt;nowiki&gt;&lt;syntaxhighlight&gt;&lt;/nowiki&gt;&lt;/code&gt; tags and on code pages (JavaScript, CSS, Scribunto and Sanitized CSS). Uses of template syntax &lt;code&gt;&lt;nowiki&gt;{{…}}&lt;/nowiki&gt;&lt;/code&gt; are also linked to the template page. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T368166]
* Two bugs were fixed in the [[m:Special:MyLanguage/Account vanishing|GlobalVanishRequest]] system by improving the logging and by removing an incorrect placeholder message. [https://phabricator.wikimedia.org/T370595][https://phabricator.wikimedia.org/T372223]
* View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] From [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]]: 
** The API now enables 5,000 on-demand API requests per month and twice-monthly HTML snapshots freely (gratis and libre). More information on the updates and also improvements to the software development kits (SDK) are explained on [https://enterprise.wikimedia.com/blog/enhanced-free-api/ the project's blog post]. While Wikimedia Enterprise APIs are designed for high-volume commercial reusers, this change enables many more community use-cases to be built on the service too.
** The Snapshot API (html dumps) have added beta Structured Contents endpoints ([https://enterprise.wikimedia.com/blog/structured-contents-snapshot-api/ blog post on that]) as well as released two beta datasets (English and French Wikipedia) from that endpoint to Hugging Face for public use and feedback ([https://enterprise.wikimedia.com/blog/hugging-face-dataset/ blog post on that]). These pre-parsed data sets enable new options for researchers, developers, and data scientists to use and study the content.

'''In depth'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The Wikidata Query Service (WDQS) is used to get answers to questions using the Wikidata data set. As Wikidata grows, we had to make a major architectural change so that WDQS could remain performant. As part of the [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split project]], we have new SPARQL endpoints available for serving the &quot;[https://query-scholarly.wikidata.org scholarly]&quot; and &quot;[https://query-main.wikidata.org main]&quot; subgraphs of Wikidata. The [http://query.wikidata.org query.wikidata.org endpoint] will continue to serve the full Wikidata graph until March 2025. After this date, it will only serve the main graph. For more information, please see [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/September 2024 scaling update|the announcement on Wikidata]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:36, 23 September 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27493779 --&gt;

== Tech News: 2024-40 ==

&lt;section begin=&quot;technews-2024-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/40|Translations]] are available.

'''Updates for editors'''
* Readers of [[phab:T375401|42 more wikis]] can now use Dark Mode. If the option is not yet available for logged-out users of your wiki, this is likely because many templates do not yet display well in Dark Mode. Please use the [https://night-mode-checker.wmcloud.org/ night-mode-checker tool] if you are interested in helping to reduce the number of issues. The [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations page]] provides guidance on this. Dark Mode is enabled on additional wikis once per month.
* Editors using the 2010 wikitext editor as their default can access features from the 2017 wikitext editor by adding &lt;code dir=ltr&gt;?veaction=editsource&lt;/code&gt; to the URL. If you would like to enable the 2017 wikitext editor as your default, it can be set in [[Special:Preferences#mw-input-wpvisualeditor-newwikitext|your preferences]]. [https://phabricator.wikimedia.org/T239796]
* For logged-out readers using the Vector 2022 skin, the &quot;donate&quot; link has been moved from a collapsible menu next to the content area into a more prominent top menu, next to &quot;Create an account&quot;. This restores the link to the level of prominence it had in the Vector 2010 skin. [[mw:Readers/2024 Reader and Donor Experiences#Donor Experiences (Key Result WE 3.2 and the related hypotheses)|Learn more]] about the changes related to donor experiences. [https://phabricator.wikimedia.org/T373585]
* The CampaignEvents extension provides tools for organizers to more easily manage events, communicate with participants, and promote their events on the wikis. The extension has been [[m:Special:MyLanguage/CampaignEvents/Deployment status|enabled]] on Arabic Wikipedia, Igbo Wikipedia, Swahili Wikipedia, and Meta-Wiki. [[w:zh:Wikipedia:互助客栈/其他#引進CampaignEvents擴充功能|Chinese Wikipedia has decided]] to enable the extension, and discussions on the extension are in progress [[w:es:Wikipedia:Votaciones/2024/Sobre la política de Organizadores de Eventos|on Spanish Wikipedia]] and [[d:Wikidata:Project chat#Enabling the CampaignEvents Extention on Wikidata|on Wikidata]]. To learn how to enable the extension on your wiki, you can visit [[m:Special:MyLanguage/CampaignEvents|the CampaignEvents page on Meta-Wiki]].
* View all {{formatnum:22}} community-submitted {{PLURAL:22|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Developers with an account on Wikitech-wiki should [[wikitech:Wikitech/SUL-migration|check if any action is required]] for their accounts. The wiki is being changed to use the single-user-login (SUL) system, and other configuration changes. This change will help reduce the overall complexity for the weekly software updates across all our wikis.

'''In depth'''
* The [[m:Special:MyLanguage/Tech/Server switch|server switch]] was completed successfully last week with a read-only time of [[wikitech:Switch Datacenter#Past Switches|only 2 minutes 46 seconds]]. This periodic process makes sure that engineers can switch data centers and keep all of the wikis available for readers, even if there are major technical issues. It also gives engineers a chance to do maintenance and upgrades on systems that normally run 24 hours a day, and often helps to reveal weaknesses in the infrastructure. The process involves dozens of software services and hundreds of hardware servers, and requires multiple teams working together. Work over the past few years has reduced the time from 17 minutes down to 2–3 minutes. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/66ZW7B2MG63AESQVTXDIFQBDBS766JGW/]

'''Meetings and events'''
* October 4–6: [[m:Special:MyLanguage/WikiIndaba conference 2024|WikiIndaba Conference's Hackathon]] in Johannesburg, South Africa
* November 4–6: [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] in Vienna, Austria

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:20, 30 September 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27530062 --&gt;

== Tech News: 2024-41 ==

&lt;section begin=&quot;technews-2024-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/41|Translations]] are available.

'''Weekly highlight'''
* Communities can now request installation of [[mw:Special:MyLanguage/Moderator Tools/Automoderator|Automoderator]] on their wiki. Automoderator is an automated anti-vandalism tool that reverts bad edits based on scores from the new &quot;Revert Risk&quot; machine learning model. You can [[mw:Special:MyLanguage/Extension:AutoModerator/Deploying|read details about the necessary steps]] for installation and configuration. [https://phabricator.wikimedia.org/T336934]

'''Updates for editors'''
* Translators in wikis where [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|the mobile experience of Content Translation is available]], can now customize their articles suggestion list from 41 filtering options when using the tool. This topic-based article suggestion feature makes it easy for translators to self-discover relevant articles based on their area of interest and translate them. You can [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;active-list=suggestions try it with your mobile device]. [https://phabricator.wikimedia.org/T368422]
* View all {{formatnum:12}} community-submitted {{PLURAL:12|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* It is now possible for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;syntaxhighlight&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; code blocks to offer readers a &quot;Copy&quot; button if the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;copy=1&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute is [[mw:Special:MyLanguage/Extension:SyntaxHighlight#copy|set on the tag]]. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T40932]
* Customized copyright footer messages on all wikis will be updated. The new versions will use wikitext markup instead of requiring editing raw HTML. [https://phabricator.wikimedia.org/T375789]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Later this month, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] will be rolled out on several pilot wikis. The final list of the wikis will be published in the second half of the month. If you maintain any tools, bots, or gadgets on [[phab:T376499|these 11 wikis]], and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]].
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Rate limiting has been enabled for the code review tools [[Wikitech:Gerrit|Gerrit]] and [[Wikitech:GitLab|GitLab]] to address ongoing issues caused by malicious traffic and scraping. Clients that open too many concurrent connections will be restricted for a few minutes. This rate limiting is managed through [[Wikitech:nftables|nftables]] firewall rules. For more details, see Wikitech's pages on [[Wikitech:Firewall#Throttling with nftables|Firewall]], [[Wikitech:GitLab/Abuse and rate limiting|GitLab limits]] and [[Wikitech:Gerrit/Operations#Throttling IPs|Gerrit operations]].
* Five new wikis have been created: 
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q49224|Komering]] ([[w:kge:|&lt;code&gt;w:kge:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374813]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q36096|Mooré]] ([[m:mos:|&lt;code&gt;m:mos:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374641]
** a {{int:project-localized-name-group-wiktionary}} in [[d:Q36213|Madurese]] ([[wikt:mad:|&lt;code&gt;wikt:mad:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374968]
** a {{int:project-localized-name-group-wikiquote}} in [[d:Q2501174|Gorontalo]] ([[q:gor:|&lt;code&gt;q:gor:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T375088]
** a {{int:project-localized-name-group-wikinews}} in [[d:Q56482|Shan]] ([[n:shn:|&lt;code&gt;n:shn:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T375430]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W41&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:42, 7 October 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27557422 --&gt;

== Tech News: 2024-42 ==

&lt;section begin=&quot;technews-2024-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/42|Translations]] are available.

'''Updates for editors'''
* The Structured Discussion extension (also known as Flow) is starting to be removed. This extension is unmaintained and causes issues. It will be replaced by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]], which is used on any regular talk page. [[mw:Special:MyLanguage/Structured Discussions/Deprecation#Deprecation timeline|A first set of wikis]] are being contacted. These wikis are invited to stop using Flow, and to move all Flow boards to sub-pages, as archives. At these wikis, a script will move all Flow pages that aren't a sub-page to a sub-page automatically, starting on 22 October 2024. On 28 October 2024, all Flow boards at these wikis will be set in read-only mode. [https://www.mediawiki.org/wiki/Structured_Discussions/Deprecation][https://phabricator.wikimedia.org/T370722]
* WMF's Search Platform team is working on making it easier for readers to perform text searches in their language. A [[phab:T332342|change last week]] on over 30 languages makes it easier to find words with accents and other diacritics. This applies to both full-text search and to types of advanced search such as the &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;''hastemplate''&lt;/bdi&gt; and &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;''incategory''&lt;/bdi&gt; keywords. More technical details (including a few other minor search upgrades) are available. [https://www.mediawiki.org/wiki/User:TJones_%28WMF%29/Notes/Language_Analyzer_Harmonization_Notes#ASCII-folding/ICU-folding_%28T332342%29]
* View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Help:Edit check|EditCheck]] was installed at Russian Wikipedia, and fixes were made for some missing user interface styles.

'''Updates for technical contributors'''
* Editors who use the Toolforge tool [[toolforge:copyvios|Earwig's Copyright Violation Detector]] will now be required to log in with their Wikimedia account before running checks using the &quot;search engine&quot; option. This change is needed to help prevent external bots from misusing the system. Thanks to Chlod for these improvements. [https://en.wikipedia.org/wiki/Wikipedia_talk:New_pages_patrol/Reviewers#Authentication_is_now_required_for_search_engine_checks_on_Earwig's_Copyvio_Tool]
* [[m:Special:MyLanguage/Phabricator|Phabricator]] users can create tickets and add comments on existing tickets via Email again. [[mw:Special:MyLanguage/Phabricator/Help#Using email|Sending email to Phabricator]] has been fixed. [https://phabricator.wikimedia.org/T356077]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Some HTML elements in the interface are now wrapped with a &lt;code&gt;&lt;nowiki&gt;&lt;bdi&gt;&lt;/nowiki&gt;&lt;/code&gt; element, to make our HTML output more aligned with Web standards. More changes like this will be coming in future weeks. This change might break some tools that rely on the previous HTML structure of the interface. Note that relying on the HTML structure of the interface is [[mw:Special:MyLanguage/Stable interface policy/Frontend#What is not stable?|not recommended]] and might break at any time. [https://phabricator.wikimedia.org/T375975]

'''In depth'''
* The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/September 2024|MediaWiki Product Insights newsletter]] is available. This edition includes: updates on Wikimedia's authentication system, research to simplify feature development in the MediaWiki platform, updates on Parser Unification and MathML rollout, and more.
* The latest quarterly [[mw:Technical Community Newsletter/2024/October|Technical Community Newsletter]] is now available. This edition include: research about improving topic suggestions related to countries, improvements to PHPUnit tests, and more.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:21, 14 October 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27597254 --&gt;

== Tech News: 2024-43 ==

&lt;section begin=&quot;technews-2024-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/43|Translations]] are available.

'''Weekly highlight'''
* The Mobile Apps team has released an [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh#Phase 1: Creating a user Profile Menu (T373714)|update]] to the iOS app's navigation, and it is now available in the latest App store version. The team added a new Profile menu that allows for easy access to editor features like Notifications and Watchlist from the Article view, and brings the &quot;Donate&quot; button into a more accessible place for users who are reading an article. This is the first phase of a larger planned [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh|navigation refresh]] to help the iOS app transition from a primarily reader-focused app, to an app that fully supports reading and editing. The Wikimedia Foundation has added more editing features and support for on-wiki communication based on volunteer requests in recent years.
[[File:IOS App Navigation refresh first phase 05.png|thumb|iOS Wikipedia App's profile menu and contents]]

'''Updates for editors'''
* Wikipedia readers can now download a browser extension to experiment with some early ideas on potential features that recommend articles for further reading, automatically summarize articles, and improve search functionality. For more details and to stay updated, check out the Web team's [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments|Content Discovery Experiments page]] and [[mw:Special:MyLanguage/Newsletter:Web team's projects|subscribe to their newsletter]].
* Later this month, logged-out editors of [[phab:T376499|these 12 wikis]] will start to have [[mw:Special:Mylanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] created. The list may slightly change - some wikis may be removed but none will be added. Temporary account is a new [[mw:Special:MyLanguage/User account types|type of user account]]. It enhances the logged-out editors' privacy and makes it easier for community members to communicate with them. If you maintain any tools, bots, or gadgets on these 12 wikis, and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]]. Read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|deployment plan across all wikis]].
* View all {{formatnum:33}} community-submitted {{PLURAL:33|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[w:nr:Main Page|South Ndebele]], [[w:rsk:Главни бок|Pannonian Rusyn]], [[w:ann:Uwu|Obolo]], [[w:iba:Lambar Keterubah|Iban]] and [[w:tdd:ᥞᥨᥝᥴ ᥘᥣᥲ ᥖᥥᥰ|Tai Nüa]] Wikipedia languages were created last week. [https://www.wikidata.org/wiki/Q36785][https://www.wikidata.org/wiki/Q35660][https://www.wikidata.org/wiki/Q36614][https://www.wikidata.org/wiki/Q33424][https://www.wikidata.org/wiki/Q36556]
* It is now possible to create functions on Wikifunctions using Wikidata lexemes, through the new [[f:Z6005|Wikidata lexeme type]] launched last week. When you go to one of these functions, the user interface provides a lexeme selector that helps you pick a lexeme from Wikidata that matches the word you type. After hitting run, your selected lexeme is retrieved from Wikidata, transformed into a Wikidata lexeme type, and passed into the selected function. Read more about this in [[f:Special:MyLanguage/Wikifunctions:Status updates/2024-10-17#Function of the Week: select representation from lexeme|the latest Wikifunctions newsletter]].

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Users of the Wikimedia sites can now format dates more easily in different languages with the new &lt;code dir=&quot;ltr&quot;&gt;{{[[mw:Special:MyLanguage/Help:Extension:ParserFunctions##timef|#timef]]:…}}&lt;/code&gt; parser function. For example, &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{#timef:now|date|en}}&lt;/nowiki&gt;&lt;/code&gt; will show as &quot;&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;{{#timef:now|date|en}}&lt;/bdi&gt;&quot;. Previously, &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{#time:…}}&lt;/nowiki&gt;&lt;/code&gt; could be used to format dates, but this required knowledge of the order of the time and date components and their intervening punctuation. &lt;code dir=&quot;ltr&quot;&gt;#timef&lt;/code&gt; (or &lt;code dir=&quot;ltr&quot;&gt;#timefl&lt;/code&gt; for local time) provides access to the standard date formats that MediaWiki uses in its user interface. This may help to simplify some templates on multi-lingual wikis like Commons and Meta. [https://phabricator.wikimedia.org/T223772][https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:ParserFunctions##timef]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Commons and Meta users can now efficiently [[mw:Special:MyLanguage/Help:Magic words#Localization|retrieve the user's language]] using &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{USERLANGUAGE}}&lt;/nowiki&gt;&lt;/code&gt; instead of using &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{int:lang}}&lt;/nowiki&gt;&lt;/code&gt;. [https://phabricator.wikimedia.org/T4085]
* The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Tech Advisory Council]] (PTAC) now has its pilot members with representation across Africa, Asia, Europe, North America and South America. They will work to address the [[Special:MyLanguage/Movement Strategy/Initiatives/Technology Council|Movement Strategy's Technology Council]] initiative of having a co-defined and more resilient technological platform. [https://meta.wikimedia.org/wiki/Movement_Strategy/Initiatives/Technology_Council]

'''In depth'''
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/32|Growth newsletter]] is available. It includes: an upcoming Newcomer Homepage Community Updates module, new Community Configuration options, and details on new projects.
* The Wikimedia Foundation is [[mw:Special:MyLanguage/Wikimedia Security Team#CNA Partnership|now an official partner of the CVE program]], which is an international effort to catalog publicly disclosed cybersecurity vulnerabilities. This partnership will allow the Security Team to instantly publish [[w:en:Common Vulnerabilities and Exposures|common vulnerabilities and exposures]] (CVE) records that are affecting MediaWiki core, extensions, and skins, along with any other code the Foundation is a steward of.
* The [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]] is now [[m:Community Wishlist/Updates#October 16, 2024: Conversations Made Easier: Machine-Translated Wishes Are Here!|testing machine translations]] for Wishlist content. Volunteers can now read machine-translated versions of wishes and dive into discussions even before translators arrive to translate content.

'''Meetings and events'''
* 24 October - Wiki Education Speaker Series Webinar - [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/N4XTB4G55BUY3M3PNGUAKQWJ7A4UOPAK/ Open Source Tech: Building the Wiki Education Dashboard], featuring Wikimedia interns and a Web developer in the panel.
* 20–22 December 2024 - [[m:Special:MyLanguage/Indic Wikimedia Hackathon Bhubaneswar 2024|Indic Wikimedia Hackathon Bhubaneswar 2024]] in Odisha, India. A hackathon for community members, including developers, designers and content editors, to build technical solutions that improve contributors' experiences.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:52, 21 October 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27634672 --&gt;

== Tech News: 2024-44 ==

&lt;section begin=&quot;technews-2024-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/44|Translations]] are available.

'''Updates for editors'''
* Later in November, the Charts extension will be deployed to the test wikis in order to help identify and fix any issue. A security review is underway to then enable deployment to pilot wikis for broader testing. You can read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates#October 2024: Working towards production deployment|the October project update]] and see the [https://en.wikipedia.beta.wmflabs.org/wiki/Charts latest documentation and examples on Beta Wikipedia].
* View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[w:en:PediaPress|Pediapress.com]], an external service that creates books from Wikipedia, can now use [[mw:Special:MyLanguage/Wikimedia Maps|Wikimedia Maps]] to include existing pre-rendered infobox map images in their printed books on Wikipedia. [https://phabricator.wikimedia.org/T375761]

'''Updates for technical contributors'''
* Wikis can use [[:mw:Special:MyLanguage/Extension:GuidedTour|the Guided Tour extension]] to help newcomers understand how to edit. The Guided Tours extension now works with [[mw:Special:MyLanguage/Manual:Dark mode|dark mode]]. Guided Tour maintainers can check their tours to see that nothing looks odd. They can also set &lt;code&gt;emitTransitionOnStep&lt;/code&gt; to &lt;code&gt;true&lt;/code&gt; to fix an old bug. They can use the new flag &lt;code&gt;allowAutomaticBack&lt;/code&gt; to avoid back-buttons they don't want. [https://phabricator.wikimedia.org/T73927#10241528]
*  Administrators in the Wikimedia projects who use the [[mw:Special:MyLanguage/Help:Extension:Nuke|Nuke Extension]] will notice that mass deletions done with this tool have the &quot;Nuke&quot; tag. This change will make reviewing and analyzing deletions performed with the tool easier. [https://phabricator.wikimedia.org/T366068]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:56, 28 October 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27668811 --&gt;

== Tech News: 2024-45 ==

&lt;section begin=&quot;technews-2024-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/45|Translations]] are available.

'''Updates for editors'''
* Stewards can now make [[m:Special:MyLanguage/Global blocks|global account blocks]] cause global [[mw:Special:MyLanguage/Autoblock|autoblocks]]. This will assist stewards in preventing abuse from users who have been globally blocked. This includes preventing globally blocked temporary accounts from exiting their session or switching browsers to make subsequent edits for 24 hours. Previously, temporary accounts could exit their current session or switch browsers to continue editing. This is an anti-abuse tool improvement for the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project. You can read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|progress on key features for temporary accounts]]. [https://phabricator.wikimedia.org/T368949]
* Wikis that have the [[m:Special:MyLanguage/CampaignEvents/Deployment status|CampaignEvents extension enabled]] can now use the [[m:Special:MyLanguage/Campaigns/Foundation Product Team/Event list#October 29, 2024: Collaboration List launched|Collaboration List]] feature. This list provides a new, easy way for contributors to learn about WikiProjects on their wikis. Thanks to the Campaign team for this work that is part of [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product %26 Technology OKRs#WE KRs|the 2024/25 annual plan]]. If you are interested in bringing the CampaignEvents extension to your wiki, you can [[m:Special:MyLanguage/CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|follow these steps]] or you can reach out to User:Udehb-WMF for help.
* The text color for red links will be slightly changed later this week to improve their contrast in light mode. [https://phabricator.wikimedia.org/T370446]
* View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, on multilingual wikis, users [[phab:T216368|can now]] hide translations from the WhatLinksHere special page.

'''Updates for technical contributors'''
* XML [[m:Special:MyLanguage/Data dumps|data dumps]] have been temporarily paused whilst a bug is investigated. [https://lists.wikimedia.org/hyperkitty/list/xmldatadumps-l@lists.wikimedia.org/message/BXWJDPO5QI2QMBCY7HO36ELDCRO6HRM4/]

'''In depth'''
* Temporary Accounts have been deployed to six wikis; thanks to the Trust and Safety Product team for [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|this work]], you can read about [[phab:T340001|the deployment plans]]. Beginning next week, Temporary Accounts will also be enabled on [[phab:T378336|seven other projects]]. If you are active on these wikis and need help migrating your tools, please reach out to [[m:User:Udehb-WMF|User:Udehb-WMF]] for assistance.
*  The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/October|Language and Internationalization newsletter]] is available.  It includes: New languages supported in translatewiki or in MediaWiki; New keyboard input methods for some languages; details about recent and upcoming meetings, and more.

'''Meetings and events'''
* [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] is happening in Vienna, Austria and online from 4 to 6 November 2024. The conference will feature discussions around the usage of MediaWiki software by and within companies in different industries and will inspire and onboard new users.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:50, 4 November 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27693917 --&gt;

== Tech News: 2024-46 ==

&lt;section begin=&quot;technews-2024-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/46|Translations]] are available.

'''Updates for editors'''
* On wikis with the [[mw:Special:MyLanguage/Help:Extension:Translate|Translate extension]] enabled, users will notice that the FuzzyBot will now automatically create translated versions of categories used on translated pages. [https://phabricator.wikimedia.org/T285463]
* View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the submitted task to use the [[mw:Special:MyLanguage/Extension:SecurePoll|SecurePoll extension]] for English Wikipedia's special [[w:en:Wikipedia:Administrator elections|administrator election]] was resolved on time. [https://phabricator.wikimedia.org/T371454] 

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] In &lt;code dir=&quot;ltr&quot;&gt;[[mw:MediaWiki_1.44/wmf.2|1.44.0-wmf-2]]&lt;/code&gt;, the logic of Wikibase function &lt;code&gt;getAllStatements&lt;/code&gt; changed to behave like &lt;code&gt;getBestStatements&lt;/code&gt;. Invoking the function now returns a copy of values which are immutable. [https://phabricator.wikimedia.org/T270851]
* [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. The API will be rerouting some page content endpoints from RESTbase to the newer [[mw:Special:MyLanguage/API:REST API|MediaWiki REST API]] endpoints. The [[phab:T374683|impacted endpoints]] include getting page/revision metadata and rendered HTML content. These changes will be available on testwiki later this week, with other projects to follow. This change should not affect existing functionality, but active users of the impacted endpoints should verify behavior on testwiki, and raise any concerns on the related [[phab:T374683|Phabricator ticket]].

'''In depth'''
* Admins and users of the Wikimedia projects [[mw:Special:MyLanguage/Moderator_Tools/Automoderator#Usage|where Automoderator is enabled]] can now monitor and evaluate important metrics related to Automoderator's actions. [https://superset.wmcloud.org/superset/dashboard/unified-automoderator-activity-dashboard/ This Superset dashboard] calculates and aggregates metrics about Automoderator's behaviour on the projects in which it is deployed. Thanks to the Moderator Tools team for this Dashboard; you can visit [[mw:Special:MyLanguage/Moderator Tools/Automoderator/Unified Activity Dashboard|the documentation page]] for more information about this work. [https://phabricator.wikimedia.org/T369488]

'''Meetings and events'''
* 21 November 2024 ([[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] &amp; [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]) - [[c:Commons:WMF support for Commons/Commons community calls|Community call]] with Wikimedia Commons volunteers and stakeholders to help prioritize support efforts for 2025-2026 Fiscal Year. The theme of this call is how content should be organised on Wikimedia Commons.
'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:07, 12 November 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27732268 --&gt;

== Tech News: 2024-47 ==

&lt;section begin=&quot;technews-2024-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/47|Translations]] are available.

'''Updates for editors'''
* Users of Wikimedia sites will now be warned when they create a [[mw:Special:MyLanguage/Help:Redirects|redirect]] to a page that doesn't exist. This will reduce the number of broken redirects to red links in our projects. [https://phabricator.wikimedia.org/T326057]
* View all {{formatnum:42}} community-submitted {{PLURAL:42|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Manual:Pywikibot/Overview|Pywikibot]], which automates work on MediaWiki sites, was upgraded to 9.5.0 on Toolforge. [https://phabricator.wikimedia.org/T378676]

'''Updates for technical contributors'''
* On wikis that use the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs extension]], pages created or moved by users with the appropriate permissions are marked as flagged automatically. This feature has not been working recently, and changes fixing it should be deployed this week. Thanks to Daniel and Wargo for working on this. [https://phabricator.wikimedia.org/T379218][https://phabricator.wikimedia.org/T368380]

'''In depth'''
* There is a new [https://diff.wikimedia.org/2024/11/05/say-hi-to-temporary-accounts-easier-collaboration-with-logged-out-editors-with-better-privacy-protection Diff post] about Temporary Accounts, available in more than 15 languages. Read it to learn about what Temporary Accounts are, their impact on different groups of users, and the plan to introduce the change on all wikis.

'''Meetings and events'''
* Technical volunteers can now register for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]], which will take place in Istanbul, Turkey. [https://pretix.eu/wikimedia/hackathon2025/ Application for travel and accommodation scholarships] is open from '''November 12 to December 10 2024'''. The registration for the event will close in mid-April 2025. The Wikimedia Hackathon is an annual gathering that unites the global technical community to collaborate on existing projects and explore new ideas.
* Join the [[C:Special:MyLanguage/Commons:WMF%20support%20for%20Commons/Commons%20community%20calls|Wikimedia Commons community calls]] this week to help prioritize support for Commons which will be planned for 2025–2026. The theme will be how content should be organised on Wikimedia Commons. This is an opportunity for volunteers who work on different things to come together and talk about what matters for the future of the project. The calls will take place '''November 21, 2024, [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] and [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]'''.
* A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place '''November 29, 16:00 UTC''' to discuss updates and technical problem-solving.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:00, 19 November 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27806858 --&gt;

== Tech News: 2024-48 ==

&lt;section begin=&quot;technews-2024-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/48|Translations]] are available.

'''Updates for editors'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new version of the standard wikitext editor-mode [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will be available as a [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] later this week. This brings many new features and bug fixes, including right-to-left support, [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Template folding|template folding]], [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Autocompletion|autocompletion]], and an improved search panel. You can learn more on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]].
* The 2010 wikitext editor now supports common keyboard shortcuts such &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;B&lt;/code&gt;&lt;/bdi&gt; for bold and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;I&lt;/code&gt;&lt;/bdi&gt; for italics. A full [[mw:Help:Extension:WikiEditor#Keyboard shortcuts|list of all six shortcuts]] is available. Thanks to SD0001 for this improvement. [https://phabricator.wikimedia.org/T62928]
* Starting November 28, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: &lt;bdi&gt;bswiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;elwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;euwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;fawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;fiwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikiquote&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikisource&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikiversity&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikivoyage&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;idwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;lvwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;plwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;ptwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;urwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;viwikisource&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;zhwikisource&lt;/bdi&gt;. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. 
* View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a user creating a new AbuseFilter can now only set the filter to &quot;protected&quot; [[phab:T377765|if it includes a protected variable]].

'''Updates for technical contributors'''
* The [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]], which can be used in JavaScript, CSS, JSON, and Lua pages, [[phab:T377663|now offers]] live autocompletion. Thanks to SD0001 for this improvement. The feature can be temporarily disabled on a page by pressing &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;,&lt;/code&gt;&lt;/bdi&gt; and un-selecting &quot;&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;Live Autocompletion&lt;/bdi&gt;&quot;.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool-maintainers who use the Graphite system for tracking metrics, need to migrate to the newer Prometheus system. They can check [https://grafana.wikimedia.org/d/K6DEOo5Ik/grafana-graphite-datasource-utilization?orgId=1 this dashboard] and the list in the Description of the [[phab:T350592|task T350592]] to see if their tools are listed, and they should claim metrics and dashboards connected to their tools. They can then disable or migrate all existing metrics by following the instructions in the task. The Graphite service will become read-only in April. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/KLUV4IOLRYXPQFWD6WKKJUHMWE77BMSZ/]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The [[mw:Special:MyLanguage/NewPP parser report|New PreProcessor parser performance report]] has been fixed to give an accurate count for the number of Wikibase entities accessed. It had previously been resetting after 400 entities. [https://phabricator.wikimedia.org/T279069]

'''Meetings and events'''
* A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place November 29 at [https://zonestamp.toolforge.org/1732896000 16:00 UTC]. There will be presentations on topics like developing language keyboards, the creation of the Mooré Wikipedia, the language support track at [[m:Wiki Indaba|Wiki Indaba]], and a report from the Wayuunaiki community on their experiences with the Incubator and as a new community over the last 3 years. This meeting will be in English and will also have Spanish interpretation.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:42, 25 November 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27847039 --&gt;

== Tech News: 2024-49 ==

&lt;section begin=&quot;technews-2024-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/49|Translations]] are available.

'''Updates for editors'''
* Two new parser functions were added this week. The &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic words#interwikilink|#interwikilink]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; function adds an [[mw:Special:MyLanguage/Help:Links#Interwiki links|interwiki link]] and the &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic words#interlanguagelink|#interlanguagelink]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; function adds an [[mw:Special:MyLanguage/Help:Links#Interlanguage links|interlanguage link]]. These parser functions are useful on wikis where namespaces conflict with interwiki prefixes. For example, links beginning with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS:&lt;/code&gt;&lt;/bdi&gt; on English Wikipedia [[phab:T363538|conflict with the &lt;code&gt;mos&lt;/code&gt; language code prefix of Mooré Wikipedia]].
* Starting this week, Wikimedia wikis no longer support connections using old RSA-based HTTPS certificates, specifically rsa-2048. This change is to improve security for all users. Some older, unsupported browser or smartphone devices will be unable to connect; Instead, they will display a connectivity error. See the [[wikitech:HTTPS/Browser_Recommendations|HTTPS Browser Recommendations page]] for more-detailed information. All modern operating systems and browsers are always able to reach Wikimedia projects. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/CTYEHVNSXUD3NFAAMG3BLZVTVQWJXJAH/]
* Starting December 16, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: &lt;bdi&gt;arwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;cawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;mediawikiwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;orwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wawiktionary&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wikidatawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;zhwiki&lt;/bdi&gt;. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. [https://phabricator.wikimedia.org/T380910]
* This month the Chart extension was deployed to production and is now available on Commons and Testwiki. With the security review complete, pilot wiki deployment is expected to start in the first week of December. You can see a working version [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details.
* View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug with the &quot;Download as PDF&quot; system was fixed. [https://phabricator.wikimedia.org/T376438]

'''Updates for technical contributors'''
* In late February, temporary accounts will be rolled out on at least 10 large wikis. This deployment will have a significant effect on the community-maintained code. This is about Toolforge tools, bots, gadgets, and user scripts that use IP address data or that are available for logged-out users. The Trust and Safety Product team wants to identify this code, monitor it, and assist in updating it ahead of the deployment to minimize disruption to workflows. The team asks technical editors and volunteer developers to help identify such tools by adding them to [[mw:Trust and Safety Product/Temporary Accounts/For developers/Impacted tools|this list]]. In addition, review the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|updated documentation]] to learn how to adjust the tools. Join the discussions on the [[mw:Talk:Trust and Safety Product/Temporary Accounts|project talk page]] or in the [[discord:channels/221049808784326656/1227616742340034722|dedicated thread]] on the [[w:Wikipedia:Discord|Wikimedia Community Discord server (in English)]] for support and to share feedback. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:22, 2 December 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27873992 --&gt;

== Tech News: 2024-50 ==

&lt;section begin=&quot;technews-2024-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/50|Translations]] are available.

'''Weekly highlight'''
* Technical documentation contributors can find updated resources, and new ways to connect with each other and the Wikimedia Technical Documentation Team, at the [[mw:Special:MyLanguage/Documentation|Documentation hub]] on MediaWiki.org. This page links to: resources for writing and improving documentation, a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;#wikimedia-techdocs&lt;/bdi&gt; IRC channel on libera.chat, a listing of past and upcoming documentation events, and ways to request a documentation consultation or review. If you have any feedback or ideas for improvements to the documentation ecosystem, please [[mw:Wikimedia Technical Documentation Team#Contact us|contact the Technical Documentation Team]].

'''Updates for editors'''
[[File:Edit Check on Desktop.png|thumb|Layout change for the Edit Check feature]]
* Later this week, [[mw:Special:MyLanguage/Edit check|Edit Check]] will be relocated to a sidebar on desktop. Edit check is the feature for new editors to help them follow policies and guidelines. This layout change creates space to present people with [[mw:Edit check#1 November 2024|new Checks]] that appear ''while'' they are typing. The [[mw:Special:MyLanguage/Edit check#Reference Check A/B Test|initial results]] show newcomers encountering Edit Check are 2.2 times more likely to publish a new content edit that includes a reference and is not reverted.
* The Chart extension, which enables editors to create data visualizations, was successfully made available on MediaWiki.org and three pilot wikis (Italian, Swedish, and Hebrew Wikipedias). You can see a working examples [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details.
* Translators in wikis where the [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|mobile experience of Content Translation is available]], can now discover articles in Wikiproject campaigns of their interest from the &quot;[https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;campaign=specialcx&amp;filter-type=automatic&amp;filter-id=collections&amp;active-list=suggestions&amp;from=es&amp;to=en All collection]&quot; category in the articles suggestion feature. Wikiproject Campaign organizers can use this feature, to help translators to discover articles of interest, by adding the &lt;code dir=ltr&gt;&lt;nowiki&gt;&lt;page-collection&gt; &lt;/page-collection&gt;&lt;/nowiki&gt;&lt;/code&gt; tag to their campaign article list page on Meta-wiki. This will make those articles discoverable in the Content Translation tool. For more detailed information on how to use the tool and tag, please refer to [[mw:Special:MyLanguage/Translation suggestions: Topic-based &amp; Community-defined lists/How to use the features|the step-by-step guide]]. [https://phabricator.wikimedia.org/T378958]
* The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, now has a [[phab:T376379#10310998|multiselect filter for namespace selection]]. This enables users to select multiple specific namespaces, instead of only one or all, when fetching pages for deletion.
* The Nuke feature also now [[phab:T364225#10371365|provides links]] to the userpage of the user whose pages were deleted, and to the pages which were not selected for deletion, after page deletions are queued. This enables easier follow-up admin-actions. Thanks to Chlod and the Moderator Tools team for both of these improvements. [https://phabricator.wikimedia.org/T364225#10371365]
* The Editing Team is working on making it easier to populate citations from archive.org using the [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|Citoid]] tool, the auto-filled citation generator. They are asking communities to add two parameters preemptively, &lt;code dir=ltr&gt;archiveUrl&lt;/code&gt; and &lt;code dir=ltr&gt;archiveDate&lt;/code&gt;, within the TemplateData for each citation template using Citoid. You can see an [https://en.wikipedia.org/w/index.php?title=Template%3ACite_web%2Fdoc&amp;diff=1261320172&amp;oldid=1260788022 example of a change in a template], and a [https://global-search.toolforge.org/?namespaces=10&amp;q=%5C%22citoid%5C%22%3A%20%5C%7B&amp;regex=1&amp;title= list of all relevant templates]. [https://phabricator.wikimedia.org/T374831]
* One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9240|Indonesian]] ([[voy:id:|&lt;code&gt;voy:id:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T380726]
* Last week, all wikis had problems serving pages to logged-in users and some logged-out users for 30–45 minutes. This was caused by a database problem, and investigation is ongoing. [https://www.wikimediastatus.net/incidents/3g2ckc7bp6l9]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug in the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add Link]] feature has been fixed. Previously, the list of sections which are excluded from Add Link was partially ignored in certain cases. [https://phabricator.wikimedia.org/T380455][https://phabricator.wikimedia.org/T380329]

'''Updates for technical contributors'''
* [[mw:Special:MyLanguage/Codex|Codex]], the design system for Wikimedia, now has an early-stage [[gitiles:design/codex-php|implementation in PHP]]. It is available for general use in MediaWiki extensions and Toolforge apps through [https://packagist.org/packages/wikimedia/codex Composer], with use in MediaWiki core coming soon. More information is available in [[wmdoc:design-codex-php/main/index.html|the documentation]]. Thanks to Doğu for the inspiration and many contributions to the library. [https://phabricator.wikimedia.org/T379662]
* [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. On December 4, the MediaWiki Interfaces team began rerouting page/revision metadata and rendered HTML content endpoints on [[testwiki:|testwiki]] from RESTbase to comparable MediaWiki REST API endpoints. The team encourages active users of these endpoints to verify their tool's behavior on testwiki and raise any concerns on the related [[phab:T374683|Phabricator ticket]] before the end of the year, as they intend to roll out the same change across all Wikimedia projects in early January. These changes are part of the work to replace the outdated [[mw:RESTBase/deprecation|RESTBase]] system.
* The [https://wikimediafoundation.limesurvey.net/986172 2024 Developer Satisfaction Survey] is seeking the opinions of the Wikimedia developer community. Please take the survey if you have any role in developing software for the Wikimedia ecosystem. The survey is open until 3 January 2025, and has an associated [[foundation:Legal:Developer Satisfaction Survey 2024 Privacy Statement|privacy statement]].
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Meetings and events'''
* The next meeting in the series of [[c:Commons:WMF support for Commons/Commons community calls|Wikimedia Foundation discussions with the Wikimedia Commons community]] will take place on [[m:Event:Commons community discussion - 12 December 2024 08:00 UTC|December 12 at 8:00 UTC]] and [[m:Event:Commons community discussion - 12_December 2024 16:00 UTC|at 16:00 UTC]]. The topic of this call is new media and new contributors. Contributors from all wikis are welcome to attend.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:16, 9 December 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27919424 --&gt;

== Tech News: 2024-51 ==

&lt;section begin=&quot;technews-2024-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/51|Translations]] are available.

'''Weekly highlight'''
* Interested in improving event management on your home wiki? The [[m:Special:MyLanguage/CampaignEvents|CampaignEvents extension]] offers organizers features like event registration management, event/wikiproject promotion, finding potential participants, and more - all directly on-wiki. If you are an organizer or think your community would benefit from this extension, start a discussion to enable it on your wiki today. To learn more about how to enable this extension on your wiki, visit the [[m:CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|deployment status page]].

'''Updates for editors'''
* Users of the iOS Wikipedia App in Italy and Mexico on the Italian, Spanish, and English Wikipedias, can see a [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Personalized Wikipedia Year in Review|personalized Year in Review]] with insights based on their reading and editing history.
* Users of the Android Wikipedia App in Sub-Saharan Africa and South Asia can see the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Rabbit Holes|Rabbit Holes]] feature. This feature shows a suggested search term in the Search bar based on the current article being viewed, and a suggested reading list generated from the user’s last two visited articles.
* The [[m:Special:MyLanguage/Global reminder bot|global reminder bot]] is now active and running on nearly 800 wikis. This service reminds most users holding temporary rights when they are about to expire, so that they can renew should they want to. See [[m:Global reminder bot/Technical details|the technical details page]] for more information.
* The next issue of Tech News will be sent out on 13 January 2025 because of the end of year holidays. Thank you to all of the translators, and people who submitted content or feedback, this year.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was [[phab:T374988|fixed]] in the Android Wikipedia App which had caused translatable SVG images to show the wrong language when they were tapped.

'''Updates for technical contributors'''
* There is no new MediaWiki version next week. The next deployments will start on 14 January. [https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar/2025] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:24, 16 December 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27942374 --&gt;

== Tech News: 2025-03 ==

&lt;section begin=&quot;technews-2025-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/03|Translations]] are available.

'''Weekly highlight'''
* The Single User Login system is being updated over the next few months. This is the system which allows users to fill out the login form on one Wikimedia site and get logged in on all others at the same time. It needs to be updated because of the ways that browsers are increasingly restricting cross-domain cookies. To accommodate these restrictions, login and account creation pages will move to a central domain, but it will still appear to the user as if they are on the originating wiki. The updated code will be enabled this week for users on test wikis. This change is planned to roll out to all users during February and March. See [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3#Deployment|the SUL3 project page]] for more details and a timeline.

'''Updates for editors'''
* On wikis with [[mw:Special:MyLanguage/Extension:PageAssessments|PageAssessments]] installed, you can now [[mw:Special:MyLanguage/Extension:PageAssessments#Search|filter search results]] to pages in a given WikiProject by using the &lt;code dir=ltr&gt;inproject:&lt;/code&gt; keyword. (These wikis: {{int:project-localized-name-arwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-enwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-enwikivoyage/en}}{{int:comma-separator/en}}{{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-huwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zhwiki/en}}) [https://phabricator.wikimedia.org/T378868] 
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q34129|Tigre]] ([[w:tig:|&lt;code&gt;w:tig:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T381377]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:35}} community-submitted {{PLURAL:35|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a bug with updating a user's edit-count after making a rollback edit, which is now fixed. [https://phabricator.wikimedia.org/T382592]

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia REST API users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. Starting the week of January 13, we will begin rerouting [[phab:T374683|some page content endpoints]] from RESTbase to the newer MediaWiki REST API endpoints for all wiki projects. This change was previously available on testwiki and should not affect existing functionality, but active users of the impacted endpoints may raise issues directly to the [[phab:project/view/6931/|MediaWiki Interfaces Team]] in Phabricator if they arise.
* Toolforge tool maintainers can now share their feedback on Toolforge UI, an initiative to provide a web platform that allows creating and managing Toolforge tools through a graphic interface, in addition to existing command-line workflows. This project aims to streamline active maintainers’ tasks, as well as make registration and deployment processes more accessible for new tool creators. The initiative is still at a very early stage, and the Cloud Services team is in the process of collecting feedback from the Toolforge community to help shape the solution to their needs. [[wikitech:Wikimedia Cloud Services team/EnhancementProposals/Toolforge UI|Read more and share your thoughts about Toolforge UI]]. 
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] For tool and library developers who use the OAuth system: The identity endpoint used for [[mw:Special:MyLanguage/OAuth/For Developers#Identifying the user|OAuth 1]] and [[mw:Special:MyLanguage/OAuth/For Developers#Identifying the user 2|OAuth 2]] returned a JSON object with an integer in its &lt;code&gt;sub&lt;/code&gt; field, which was incorrect (the field must always be a string). This has been fixed; the fix will be deployed to Wikimedia wikis on the week of January 13. [https://phabricator.wikimedia.org/T382139]
* Many wikis currently use [[:mw:Parsoid/Parser Unification/Cite CSS|Cite CSS]] to render custom footnote markers in Parsoid output. Starting January 20 these rules will be disabled, but the developers ask you to ''not'' clean up your &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Common.css]]&lt;/bdi&gt; until February 20 to avoid issues during the migration. Your wikis might experience some small changes to footnote markers in Visual Editor and when using experimental Parsoid read mode, but if there are changes these are expected to bring the rendering in line with the legacy parser output. [https://phabricator.wikimedia.org/T370027]

'''Meetings and events'''
* The next meeting in the series of [[c:Special:MyLanguage/Commons:WMF support for Commons/Commons community calls|Wikimedia Foundation Community Conversations with the Wikimedia Commons community]] will take place on [[m:Special:MyLanguage/Event:Commons community discussion - 15 January 2025 08:00 UTC|January 15 at 8:00 UTC]] and [[m:Special:MyLanguage/Event:Commons community discussion - 15 January 2025 16:00 UTC|at 16:00 UTC]]. The topic of this call is defining the priorities in tool investment for Commons. Contributors from all wikis, especially users who are maintaining tools for Commons, are welcome to attend.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:42, 14 January 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28048614 --&gt;

== Tech News: 2025-04 ==

&lt;section begin=&quot;technews-2025-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/04|Translations]] are available.

'''Updates for editors'''
* Administrators can mass-delete multiple pages created by a user or IP address using [[mw:Special:MyLanguage/Extension:Nuke|Extension:Nuke]]. It previously only allowed deletion of pages created in the last 30 days. It can now delete pages from the last 90 days, provided it is targeting a specific user or IP address. [https://phabricator.wikimedia.org/T380846]
* On [[phab:P72148|wikis that use]] the [[mw:Special:MyLanguage/Help:Patrolled edits|Patrolled edits]] feature, when the rollback feature is used to revert an unpatrolled page revision, that revision will now be marked as &quot;manually patrolled&quot; instead of &quot;autopatrolled&quot;, which is more accurate. Some editors that use [[mw:Special:MyLanguage/Help:New filters for edit review/Filtering|filters]] on Recent Changes may need to update their filter settings. [https://phabricator.wikimedia.org/T302140]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the Visual Editor's &quot;Insert link&quot; feature did not always suggest existing pages properly when an editor started typing, which has now been [[phab:T383497|fixed]].

'''Updates for technical contributors'''
* The Structured Discussion extension (also known as Flow) is being progressively removed from the wikis. This extension is unmaintained and causes issues. It will be replaced by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]], which is used on any regular talk page. [[mw:Special:MyLanguage/Structured Discussions/Deprecation#Deprecation timeline|The last group of wikis]] ({{int:project-localized-name-cawikiquote/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiwikimedia/en}}{{int:comma-separator/en}}{{int:project-localized-name-gomwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kabwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwikibooks/en}}{{int:comma-separator/en}}{{int:project-localized-name-sewikimedia/en}}) will soon be contacted. If you have questions about this process, please ping [[m:User:Trizek (WMF)|Trizek (WMF)]] at your wiki. [https://phabricator.wikimedia.org/T380912]
* The latest quarterly [[mw:Technical_Community_Newsletter/2025/January|Technical Community Newsletter]] is now available. This edition includes: updates about services from the Data Platform Engineering teams, information about Codex from the Design System team, and more.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:36, 21 January 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28129769 --&gt;

== Tech News: 2025-05 ==

&lt;section begin=&quot;technews-2025-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/05|Translations]] are available.

'''Weekly highlight'''
* Patrollers and admins - what information or context about edits or users could help you to make patroller or admin decisions more quickly or easily? The Wikimedia Foundation wants to hear from you to help guide its upcoming annual plan. Please consider sharing your thoughts on this and [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Product &amp; Technology OKRs|13 other questions]] to shape the technical direction for next year.

'''Updates for editors'''
* iOS Wikipedia App users worldwide can now access a [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Personalized Wikipedia Year in Review/How your data is used|personalized Year in Review]] feature, which provides insights based on their reading and editing history on Wikipedia. This project is part of a broader effort to help welcome new readers as they discover and interact with encyclopedic content.
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Edit patrollers now have a new feature available that can highlight potentially problematic new pages. When a page is created with the same title as a page which was previously deleted, a tag ('Recreated') will now be added, which users can filter for in [[{{#special:RecentChanges}}]] and [[{{#special:NewPages}}]]. [https://phabricator.wikimedia.org/T56145]
* Later this week, there will be a new warning for editors if they attempt to create a redirect that links to another redirect (a [[mw:Special:MyLanguage/Help:Redirects#Double redirects|double redirect]]). The feature will recommend that they link directly to the second redirect's target page. Thanks to the user SomeRandomDeveloper for this improvement. [https://phabricator.wikimedia.org/T326056]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia wikis allow [[w:en:WebAuthn|WebAuthn]]-based second factor checks (such as hardware tokens) during login, but the feature is [[m:Community Wishlist Survey 2023/Miscellaneous/Fix security key (WebAuthn) support|fragile]] and has very few users. The MediaWiki Platform team is temporarily disabling adding new WebAuthn keys, to avoid interfering with the rollout of [[mw:MediaWiki Platform Team/SUL3|SUL3]] (single user login version 3). Existing keys are unaffected. [https://phabricator.wikimedia.org/T378402]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''

* For developers that use the [[wikitech:Data Platform/Data Lake/Edits/MediaWiki history dumps|MediaWiki History dumps]]: The Data Platform Engineering team has added a couple of new fields to these dumps, to support the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] initiative. If you maintain software that reads those dumps, please review your code and the updated documentation, since the order of the fields in the row will change. There will also be one field rename: in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mediawiki_user_history&lt;/code&gt;&lt;/bdi&gt; dump, the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;anonymous&lt;/code&gt;&lt;/bdi&gt; field will be renamed to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;is_anonymous&lt;/code&gt;&lt;/bdi&gt;. The changes will take effect with the next release of the dumps in February. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LKMFDS62TXGDN6L56F4ABXYLN7CSCQDI/] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:14, 27 January 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28149374 --&gt;

== Tech News: 2025-06 ==

&lt;section begin=&quot;technews-2025-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/06|Translations]] are available.

'''Updates for editors'''
* Editors who use the &quot;Special characters&quot; editing-toolbar menu can now see the 32 special characters you have used most recently, across editing sessions on that wiki. This change should help make it easier to find the characters you use most often. The feature is in both the 2010 wikitext editor and VisualEditor. [https://phabricator.wikimedia.org/T110722]
* Editors using the 2010 wikitext editor can now create sublists with correct indentation by selecting the line(s) you want to indent and then clicking the toolbar buttons.[https://phabricator.wikimedia.org/T380438] You can now also insert &lt;code&gt;&lt;nowiki&gt;&lt;code&gt;&lt;/nowiki&gt;&lt;/code&gt; tags using a new toolbar button.[https://phabricator.wikimedia.org/T383010] Thanks to user stjn for these improvements.
* Help is needed to ensure the [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|citation generator]] works properly on each wiki.
** (1) Administrators should update the local versions of the page &lt;code dir=ltr&gt;MediaWiki:Citoid-template-type-map.json&lt;/code&gt; to include entries for &lt;code dir=ltr&gt;preprint&lt;/code&gt;, &lt;code dir=ltr&gt;standard&lt;/code&gt;, and &lt;code dir=ltr&gt;dataset&lt;/code&gt;; Here are example diffs to replicate [https://en.wikipedia.org/w/index.php?title=MediaWiki%3ACitoid-template-type-map.json&amp;diff=1189164774&amp;oldid=1165783565 for 'preprint'] and [https://en.wikipedia.org/w/index.php?title=MediaWiki%3ACitoid-template-type-map.json&amp;diff=1270832208&amp;oldid=1270828390 for 'standard' and 'dataset'].
** (2.1) If the citoid map in the citation template used for these types of references is missing, [[mediawikiwiki:Citoid/Enabling Citoid on your wiki#Step 2.a: Create a 'citoid' maps value for each citation template|one will need to be added]]. (2.2) If the citoid map does exist, the TemplateData will need to be updated to include new field names. Here are example updates [https://en.wikipedia.org/w/index.php?title=Template%3ACitation%2Fdoc&amp;diff=1270829051&amp;oldid=1262470053 for 'preprint'] and [https://en.wikipedia.org/w/index.php?title=Template%3ACitation%2Fdoc&amp;diff=1270831369&amp;oldid=1270829480 for 'standard' and 'dataset']. The new fields that may need to be supported are &lt;code dir=ltr&gt;archiveID&lt;/code&gt;, &lt;code dir=ltr&gt;identifier&lt;/code&gt;, &lt;code dir=ltr&gt;repository&lt;/code&gt;, &lt;code dir=ltr&gt;organization&lt;/code&gt;, &lt;code dir=ltr&gt;repositoryLocation&lt;/code&gt;, &lt;code dir=ltr&gt;committee&lt;/code&gt;, and &lt;code dir=ltr&gt;versionNumber&lt;/code&gt;. [https://phabricator.wikimedia.org/T383666]
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q15637215|Central Kanuri]] ([[w:knc:|&lt;code&gt;w:knc:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T385181]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[mediawikiwiki:Special:MyLanguage/Help:Extension:Wikisource/Wikimedia OCR|OCR (optical character recognition) tool]] used for Wikisource now supports a new language, Church Slavonic. [https://phabricator.wikimedia.org/T384782]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:09, 4 February 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28203495 --&gt;

== Tech News: 2025-07 ==

&lt;section begin=&quot;technews-2025-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/07|Translations]] are available.

'''Weekly highlight'''

* The Product and Technology Advisory Council (PTAC) has published [[m:Special:MyLanguage/Product and Technology Advisory Council/February 2025 draft PTAC recommendation for feedback|a draft of their recommendations]] for the Wikimedia Foundation's Product and Technology department. They have recommended focusing on [[m:Special:MyLanguage/Product and Technology Advisory Council/February 2025 draft PTAC recommendation for feedback/Mobile experiences|mobile experiences]], particularly contributions. They request community [[m:Talk:Product and Technology Advisory Council/February 2025 draft PTAC recommendation for feedback|feedback at the talk page]] by 21 February.

'''Updates for editors'''
* The &quot;Special pages&quot; portlet link will be moved from the &quot;Toolbox&quot; into the &quot;Navigation&quot; section of the main menu's sidebar by default. This change is because the Toolbox is intended for tools relating to the current page, not tools relating to the site, so the link will be more logically and consistently located. To modify this behavior and update CSS styling, administrators can follow the instructions at [[phab:T385346|T385346]]. [https://phabricator.wikimedia.org/T333211]
* As part of this year's work around improving the ways readers discover content on the wikis, the Web team will be running an experiment with a small number of readers that displays some suggestions for related or interesting articles within the search bar. Please check out [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments#Experiment 1: Display article recommendations in more prominent locations, search|the project page]] for more information.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Template editors who use TemplateStyles can now customize output for users with specific accessibility needs by using accessibility related media queries (&lt;code dir=ltr&gt;[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/prefers-reduced-motion prefers-reduced-motion]&lt;/code&gt;, &lt;code dir=ltr&gt;[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/prefers-reduced-transparency prefers-reduced-transparency]&lt;/code&gt;, &lt;code dir=ltr&gt;[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/prefers-contrast prefers-contrast]&lt;/code&gt;, and &lt;code dir=ltr&gt;[https://developer.mozilla.org/en-US/docs/Web/CSS/@media/forced-colors forced-colors]&lt;/code&gt;). Thanks to user Bawolff for these improvements. [https://phabricator.wikimedia.org/T384175]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:22}} community-submitted {{PLURAL:22|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the global blocks log will now be shown directly on the {{#special:CentralAuth}} page, similarly to global locks, to simplify the workflows for stewards. [https://phabricator.wikimedia.org/T377024]

'''Updates for technical contributors'''
* Wikidata [[d:Special:MyLanguage/Help:Default values for labels and aliases|now supports a special language as a &quot;default for all languages&quot;]] for labels and aliases. This is to avoid excessive duplication of the same information across many languages. If your Wikidata queries use labels, you may need to update them as some existing labels are getting removed. [https://phabricator.wikimedia.org/T312511]
* The function &lt;code dir=&quot;ltr&quot;&gt;getDescription&lt;/code&gt; was invoked on every Wiki page read and accounts for ~2.5% of a page's total load time. The calculated value will now be cached, reducing load on Wikimedia servers. [https://phabricator.wikimedia.org/T383660]
* As part of the RESTBase deprecation [[mw:RESTBase/deprecation|effort]], the &lt;code dir=&quot;ltr&quot;&gt;/page/related&lt;/code&gt; endpoint has been blocked as of February 6, 2025, and will be removed soon. This timeline was chosen to align with the deprecation schedules for older Android and iOS versions. The stable alternative is the &quot;&lt;code dir=&quot;ltr&quot;&gt;morelike&lt;/code&gt;&quot; action API in MediaWiki, and [[gerrit:c/mediawiki/services/mobileapps/+/982154/13/pagelib/src/transform/FooterReadMore.js|a migration example]] is available. The MediaWiki Interfaces team [[phab:T376297|can be contacted]] for any questions. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/GFC2IJO7L4BWO3YTM7C5HF4MCCBE2RJ2/]

'''In depth'''

* The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/January|Language and Internationalization newsletter]] is available. It includes: Updates about the &quot;Contribute&quot; menu; details on some of the newest language editions of Wikipedia; details on new languages supported by the MediaWiki interface; updates on the Community-defined lists feature; and more. 
* The latest [[mw:Extension:Chart/Project/Updates#January 2025: Better visibility into charts and tabular data usage|Chart Project newsletter]] is available. It includes updates on the progress towards bringing better visibility into global charts usage and support for categorizing pages in the Data namespace on Commons.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:12, 11 February 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28231022 --&gt;

== Tech News: 2025-08 ==

&lt;section begin=&quot;technews-2025-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/08|Translations]] are available.

'''Weekly highlight'''
* Communities using growth tools can now showcase one event on the &lt;code&gt;{{#special:Homepage}}&lt;/code&gt; for newcomers. This feature will help newcomers to be informed about editing activities they can participate in. Administrators can create a new event to showcase at &lt;code&gt;{{#special:CommunityConfiguration}}&lt;/code&gt;. To learn more about this feature, please read [[diffblog:2025/02/12/community-updates-module-connecting-newcomers-to-your-initiatives/|the Diff post]], have a look [[mw:Special:MyLanguage/Help:Growth/Tools/Community updates module|at the documentation]], or contact [[mw:Talk:Growth|the Growth team]].

'''Updates for editors'''
[[File:Page Frame Features on desktop.png|thumb|Highlighted talk pages improvements]]
* Starting next week, talk pages at these wikis – {{int:project-localized-name-eswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jawiki/en}} – will get [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|a new design]]. This change was extensively tested as a Beta feature and is the last step of [[mw:Special:MyLanguage/Talk pages project/Feature summary|talk pages improvements]]. [https://phabricator.wikimedia.org/T379102]
* You can now navigate to view a redirect page directly from its action pages, such as the history page. Previously, you were forced to first go to the redirect target. This change should help editors who work with redirects a lot. Thanks to user stjn for this improvement. [https://phabricator.wikimedia.org/T5324]
* When a Cite reference is reused many times, wikis currently show either numbers like &quot;1.23&quot; or localized alphabetic markers like &quot;a b c&quot; in the reference list. Previously, if there were so many reuses that the alphabetic markers were all used, [[MediaWiki:Cite error references no backlink label|an error message]] was displayed.  As part of the work to [[phab:T383036|modernize Cite customization]], these errors will no longer be shown and instead the backlinks will fall back to showing numeric markers like &quot;1.23&quot; once the alphabetic markers are all used.
* The log entries for each change to an editor's user-groups are now clearer by specifying exactly what has changed, instead of the plain before and after listings. Translators can [[phab:T369466|help to update the localized versions]]. Thanks to user Msz2001 for these improvements.
* A new filter has been added to the [[{{#special:Nuke}}]] tool, which allows administrators to mass delete pages, to enable users to filter for pages in a range of page sizes (in bytes). This allows, for example, deleting pages only of a certain size or below. [https://phabricator.wikimedia.org/T378488]
* Non-administrators can now check which pages are able to be deleted using the [[{{#special:Nuke}}]] tool. Thanks to user MolecularPilot for this and the previous improvements. [https://phabricator.wikimedia.org/T376378]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed in the configuration for the AV1 video file format, which enables these files to play again. [https://phabricator.wikimedia.org/T382193]

'''Updates for technical contributors'''
* Parsoid Read Views is going to be rolling out to most Wiktionaries over the next few weeks, following the successful transition of Wikivoyage to Parsoid Read Views last year. For more information, see the [[mw:Special:MyLanguage/Parsoid/Parser Unification|Parsoid/Parser Unification]] project page. [https://phabricator.wikimedia.org/T385923][https://phabricator.wikimedia.org/T371640]
* Developers of tools that run on-wiki should note that &lt;code dir=ltr&gt;mw.Uri&lt;/code&gt; is deprecated. Tools requiring &lt;code dir=ltr&gt;mw.Uri&lt;/code&gt; must explicitly declare &lt;code dir=ltr&gt;mediawiki.Uri&lt;/code&gt; as a ResourceLoader dependency, and should migrate to the browser native &lt;code dir=ltr&gt;URL&lt;/code&gt; API soon. [https://phabricator.wikimedia.org/T384515]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:16, 17 February 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28275610 --&gt;

== Tech News: 2025-09 ==

&lt;section begin=&quot;technews-2025-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/09|Translations]] are available.

'''Updates for editors'''
* Administrators can now customize how the [[m:Special:MyLanguage/User language|Babel feature]] creates categories using [[{{#special:CommunityConfiguration/Babel}}]]. They can rename language categories, choose whether they should be auto-created, and adjust other settings. [https://phabricator.wikimedia.org/T374348]
* The &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikimedia.org/ wikimedia.org]&lt;/bdi&gt; portal has been updated – and is receiving some ongoing improvements – to modernize and improve the accessibility of our portal pages. It now has better support for mobile layouts, updated wording and links, and better language support. Additionally, all of the Wikimedia project portals, such as &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://wikibooks.org wikibooks.org]&lt;/bdi&gt;, now support dark mode when a reader is using that system setting. [https://phabricator.wikimedia.org/T373204][https://phabricator.wikimedia.org/T368221][https://meta.wikimedia.org/wiki/Project_portals]
* One new wiki has been created: a {{int:project-localized-name-group-wiktionary/en}} in [[d:Q33965|Santali]] ([[wikt:sat:|&lt;code&gt;wikt:sat:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T386619]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that prevented clicking on search results in the web-interface for some Firefox for Android phone configurations. [https://phabricator.wikimedia.org/T381289]

'''Meetings and events'''
* The next Language Community Meeting is happening soon, February 28th at [https://zonestamp.toolforge.org/1740751200 14:00 UTC]. This week's meeting will cover: highlights and technical updates on keyboard and tools for the Sámi languages, Translatewiki.net contributions from the Bahasa Lampung community in Indonesia, and technical Q&amp;A. If you'd like to join, simply [[mw:Wikimedia Language and Product Localization/Community meetings#28 February 2025|sign up on the wiki page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:41, 25 February 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28296129 --&gt;

== Tech News: 2025-10 ==

&lt;section begin=&quot;technews-2025-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/10|Translations]] are available.

'''Updates for editors'''
* All logged-in editors using the mobile view can now edit a full page. The &quot;{{int:Minerva-page-actions-editfull}}&quot; link is accessible from the &quot;{{int:minerva-page-actions-overflow}}&quot; menu in the toolbar. This was previously only available to editors using the [[mw:Special:MyLanguage/Reading/Web/Advanced mobile contributions|Advanced mobile contributions]] setting. [https://phabricator.wikimedia.org/T387180]
* Interface administrators can now help to remove the deprecated Cite CSS code matching &quot;&lt;code dir=&quot;ltr&quot;&gt;mw-ref&lt;/code&gt;&quot; from their local &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Common.css]]&lt;/bdi&gt;. The list of wikis in need of cleanup, and the code to remove, [https://global-search.toolforge.org/?q=mw-ref%5B%5E-a-z%5D&amp;regex=1&amp;namespaces=8&amp;title=.*css can be found with this global search] and in [https://ace.wikipedia.org/w/index.php?title=MediaWiki:Common.css&amp;oldid=145662#L-139--L-144 this example], and you can learn more about how to help on the [[mw:Parsoid/Parser Unification/Cite CSS|CSS migration project page]]. The Cite footnote markers (&quot;&lt;code dir=&quot;ltr&quot;&gt;[1]&lt;/code&gt;&quot;) are now rendered by [[mw:Special:MyLanguage/Parsoid|Parsoid]], and the deprecated CSS is no longer needed. The CSS for backlinks (&quot;&lt;code dir=&quot;ltr&quot;&gt;mw:referencedBy&lt;/code&gt;&quot;) should remain in place for now. This cleanup is expected to cause no visible changes for readers. Please help to remove this code before March 20, after which the development team will do it for you.
* When editors embed a file (e.g. &lt;code&gt;&lt;nowiki&gt;[[File:MediaWiki.png]]&lt;/nowiki&gt;&lt;/code&gt;) on a page that is protected with cascading protection, the software will no longer restrict edits to the file description page, only to new file uploads.[https://phabricator.wikimedia.org/T24521] In contrast, transcluding a file description page (e.g. &lt;code&gt;&lt;nowiki&gt;{{:File:MediaWiki.png}}&lt;/nowiki&gt;&lt;/code&gt;) will now restrict edits to the page.[https://phabricator.wikimedia.org/T62109]
* When editors revert a file to an earlier version it will now require the same permissions as ordinarily uploading a new version of the file. The software now checks for 'reupload' or 'reupload-own' rights,[https://phabricator.wikimedia.org/T304474] and respects cascading protection.[https://phabricator.wikimedia.org/T140010]
* When administrators are listing pages for deletion with the Nuke tool, they can now also list associated talk pages and redirects for deletion, alongside pages created by the target, rather than needing to manually delete these pages afterwards. [https://phabricator.wikimedia.org/T95797]
* The [[m:Special:MyLanguage/Tech/News/2025/03|previously noted]] update to Single User Login, which will accommodate browser restrictions on cross-domain cookies by moving login and account creation to a central domain, will now roll out to all users during March and April. The team plans to enable it for all new account creation on [[wikitech:Deployments/Train#Tuesday|Group0]] wikis this week. See [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3#Deployment|the SUL3 project page]] for more details and an updated timeline.
* Since last week there has been a bug that shows some interface icons as black squares until the page has fully loaded. It will be fixed this week. [https://phabricator.wikimedia.org/T387351]
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q2044560|Sylheti]] ([[w:syl:|&lt;code&gt;w:syl:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T386441]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed with loading images in very old versions of the Firefox browser on mobile. [https://phabricator.wikimedia.org/T386400]

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.19|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:30, 4 March 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28334563 --&gt;

== Tech News: 2025-11 ==

&lt;section begin=&quot;technews-2025-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/11|Translations]] are available.

'''Updates for editors'''
* Editors who use password managers at multiple wikis may notice changes in the future. The way that our wikis provide information to password managers about reusing passwords across domains has recently been updated, so some password managers might now offer you login credentials that you saved for a different Wikimedia site. Some password managers already did this, and are now doing it for more Wikimedia domains. This is part of the [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3|SUL3 project]] which aims to improve how our unified login works, and to keep it compatible with ongoing changes to the web-browsers we use. [https://phabricator.wikimedia.org/T385520][https://phabricator.wikimedia.org/T384844]
* The Wikipedia Apps Team is inviting interested users to help improve Wikipedia’s offline and limited internet use. After discussions in [[m:Afrika Baraza|Afrika Baraza]] and the last [[m:Special:MyLanguage/ESEAP Hub/Meetings|ESEAP call]], key challenges like search, editing, and offline access are being explored, with upcoming focus groups to dive deeper into these topics. All languages are welcome, and interpretation will be available. Want to share your thoughts? [[mw:Special:MyLanguage/Wikimedia Apps/Improving Wikipedia Mobile Apps for Offline &amp; Limited Internet Use|Join the discussion]] or email &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;aramadan@wikimedia.org&lt;/bdi&gt;!
* All wikis will be read-only for a few minutes on March 19. This is planned at [https://zonestamp.toolforge.org/1742392800 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.20|MediaWiki]]

'''In depth'''
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/33|Growth newsletter]] is available. It includes: the launch of the Community Updates module, the most recent changes in Community Configuration, and the upcoming test of in-article suggestions for first-time editors.
* An old API that was previously used in the Android Wikipedia app is being removed at the end of March. There are no current software uses, but users of the app with a version that is older than 6 months by the time of removal (2025-03-31), will no longer have access to the Suggested Edits feature, until they update their app. You can [[diffblog:2025/02/24/sunset-of-wikimedia-recommendation-api/|read more details about this change]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:09, 10 March 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28372257 --&gt;

== Tech News: 2025-12 ==

&lt;section begin=&quot;technews-2025-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/12|Translations]] are available.

'''Weekly highlight'''
* Twice a year, around the equinoxes, the Wikimedia Foundation's Site Reliability Engineering (SRE) team performs [[m:Special:MyLanguage/Tech/Server switch|a datacenter server switchover]], redirecting all traffic from one primary server to its backup. This provides reliability in case of a crisis, as we can always fall back on the other datacenter. [http://listen.hatnote.com/ Thanks to the Listen to Wikipedia] tool, you can hear the switchover take place: Before it begins, you'll hear the steady stream of edits; Then, as the system enters a brief read-only phase, the sound stops for a couple of minutes, before resuming after the switchover. You can [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|read more about the background and details of this process on the Diff blog]]. If you want to keep an ear out for the next server switchover, listen to the wikis on [https://zonestamp.toolforge.org/1742392800 March 19 at 14:00 UTC].

'''Updates for editors'''

* The [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;filter-type=automatic&amp;filter-id=previous-edits&amp;active-list=suggestions&amp;from=en&amp;to=es improved Content Translation tool dashboard] is now available in [[phab:T387820|10 Wikipedias]] and will be available for all Wikipedias [[phab:T387821|soon]]. With [[mw:Special:MyLanguage/Content translation#Improved translation experience|the unified dashboard]], desktop users can now: Translate new sections of an article; Discover and access topic-based [https://ig.m.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;active-list=suggestions&amp;from=en&amp;to=ig&amp;filter-type=automatic&amp;filter-id=previous-edits article suggestion filters] (initially available only for mobile device users); Discover and access the [[mw:Special:MyLanguage/Translation suggestions: Topic-based &amp; Community-defined lists|Community-defined lists]] filter, also known as &quot;Collections&quot;, from wiki-projects and campaigns.
* On Wikimedia Commons, a [[c:Commons:WMF support for Commons/Upload Wizard Improvements#Improve category selection|new system to select the appropriate file categories]] has been introduced: if a category has one or more subcategories, users will be able to click on an arrow that will open the subcategories directly within the form, and choose the correct one. The parent category name will always be shown on top, and it will always be possible to come back to it. This should decrease the amount of work for volunteers in fixing/creating new categories. The change is also available on mobile. These changes are part of planned improvements to the UploadWizard.
* The Community Tech team is seeking wikis to join a pilot for the [[m:Special:MyLanguage/Community Wishlist Survey 2023/Multiblocks|Multiblocks]] feature and a refreshed Special:Block page in late March. Multiblocks enables administrators to impose multiple different types of blocks on the same user at the same time. If you are an admin or steward and would like us to discuss joining the pilot with your community, please leave a message on the [[m:Talk:Community Wishlist Survey 2023/Multiblocks|project talk page]].
* Starting March 25, the Editing team will test a new feature for Edit Check at [[phab:T384372|12 Wikipedias]]: [[mw:Special:MyLanguage/Help:Edit check#Multi-check|Multi-Check]]. Half of the newcomers on these wikis will see all [[mw:Special:MyLanguage/Help:Edit check#ref|Reference Checks]] during their edit session, while the other half will continue seeing only one. The goal of this test is to see if users are confused or discouraged when shown multiple Reference Checks (when relevant) within a single editing session. At these wikis, the tags used on edits that show References Check will be simplified, as multiple tags could be shown within a single edit. Changes to the tags are documented [[phab:T373949|on Phabricator]]. [https://phabricator.wikimedia.org/T379131]
* The [[m:Special:MyLanguage/Global reminder bot|Global reminder bot]], which is a service for notifying users that their temporary user-rights are about to expire, now supports using the localized name of the user-rights group in the message heading. Translators can see the [[m:Global reminder bot/Translation|listing of existing translations and documentation]] to check if their language needs updating or creation.
* The [[Special:GlobalPreferences|GlobalPreferences]] gender setting, which is used for how the software should refer to you in interface messages, now works as expected by overriding the local defaults. [https://phabricator.wikimedia.org/T386584]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:26}} community-submitted {{PLURAL:26|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the Wikipedia App for Android had a bug fixed for when a user is browsing and searching in multiple languages. [https://phabricator.wikimedia.org/T379777]

'''Updates for technical contributors'''
* Later this week, the way that Codex styles are loaded will be changing. There is a small risk that this may result in unstyled interface message boxes on certain pages. User generated content (e.g. templates) is not impacted. Gadgets may be impacted. If you see any issues [[phab:T388847|please report them]]. See the linked task for details, screenshots, and documentation on how to fix any affected gadgets.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.21|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:48, 17 March 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28412594 --&gt;

== Tech News: 2025-13 ==

&lt;section begin=&quot;technews-2025-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/13|Translations]] are available.

'''Weekly highlight'''
* The Wikimedia Foundation is seeking your feedback on the [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Product &amp; Technology OKRs|drafts of the objectives and key results that will shape the Foundation's Product and Technology priorities]] for the next fiscal year (starting in July). The objectives are broad high-level areas, and the key-results are measurable ways to track the success of their objectives. Please share your feedback on the talkpage, in any language, ideally before the end of April.

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] will be released to multiple wikis (see [[m:Special:MyLanguage/CampaignEvents/Deployment status#Global Deployment Plan|deployment plan]] for details) in April 2025, and the team has begun the process of engaging communities on the identified wikis. The extension provides tools to organize, manage, and promote collaborative activities (like events, edit-a-thons, and WikiProjects) on the wikis. The extension has three tools: [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], [[m:Special:MyLanguage/CampaignEvents/Collaboration list|Collaboration List]], and [[m:Special:MyLanguage/Campaigns/Foundation Product Team/Invitation list|Invitation Lists]]. It is currently on 13 Wikipedias, including English Wikipedia, French Wikipedia, and Spanish Wikipedia, as well as Wikidata. Questions or requests can be directed to the [[mw:Help talk:Extension:CampaignEvents|extension talk page]] or in Phabricator (with &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; style=&quot;white-space: nowrap;&quot;&gt;#campaigns-product-team&lt;/bdi&gt; tag).
* Starting the week of March 31st, wikis will be able to set which user groups can view private registrants in [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], as part of the [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents]] extension. By default, event organizers and the local wiki admins will be able to see private registrants. This is a change from the current behavior, in which only event organizers can see private registrants. Wikis can change the default setup by [[m:Special:MyLanguage/Requesting wiki configuration changes|requesting a configuration change]] in Phabricator (and adding the &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; style=&quot;white-space: nowrap;&quot;&gt;#campaigns-product-team&lt;/bdi&gt; tag). Participants of past events can cancel their registration at any time.
* Administrators at wikis that have a customized &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Sidebar]]&lt;/bdi&gt; should check that it contains an entry for the {{int:specialpages}} listing. If it does not, they should add it using &lt;code dir=ltr style=&quot;white-space: nowrap;&quot;&gt;* specialpages-url|specialpages&lt;/code&gt;. Wikis with a default sidebar will see the link moved from the page toolbox into the sidebar menu in April. [https://phabricator.wikimedia.org/T388927]
* The Minerva skin (mobile web) combines both Notice and Alert notifications within the bell icon ([[File:OOjs UI icon bell.svg|16px|link=|class=skin-invert]]). There was a long-standing bug where an indication for new notifications was only shown if you had unseen Alerts. This bug is now fixed. In the future, Minerva users will notice a counter atop the bell icon when you have 1 or more unseen Notices and/or Alerts. [https://phabricator.wikimedia.org/T344029]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* VisualEditor has introduced a [[mw:VisualEditor/Hooks|new client-side hook]] for developers to use when integrating with the VisualEditor target lifecycle. This hook should replace the existing lifecycle-related hooks, and be more consistent between different platforms. In addition, the new hook will apply to uses of VisualEditor outside of just full article editing, allowing gadgets to interact with the editor in DiscussionTools as well. The Editing Team intends to deprecate and eventually remove the old lifecycle hooks, so any use cases that this new hook does not cover would be of interest to them and can be [[phab:T355555|shared in the task]].
* Developers who use the &lt;code dir=ltr&gt;mw.Api&lt;/code&gt; JavaScript library, can now identify the tool using it with the &lt;code dir=ltr&gt;userAgent&lt;/code&gt; parameter: &lt;code dir=ltr&gt;var api = new mw.Api( { userAgent: 'GadgetNameHere/1.0.1' } );&lt;/code&gt;. If you maintain a gadget or user script, please set a user agent, because it helps with library and server maintenance and with differentiating between legitimate and illegitimate traffic. [https://phabricator.wikimedia.org/T373874][https://foundation.wikimedia.org/wiki/Policy:Wikimedia_Foundation_User-Agent_Policy]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.22|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:42, 24 March 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28443127 --&gt;

== Tech News: 2025-14 ==

&lt;section begin=&quot;technews-2025-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/14|Translations]] are available.

'''Updates for editors'''
* The Editing team is working on a new [[mw:Special:MyLanguage/Edit Check|Edit check]]: [[mw:Special:MyLanguage/Edit check#26 March 2025|Peacock check]]. This check's goal is to identify non-neutral terms while a user is editing a wikipage, so that they can be informed that their edit should perhaps be changed before they publish it. This project is at the early stages, and the team is looking for communities' input: [[phab:T389445|in this Phabricator task]], they are gathering on-wiki policies, templates used to tag non-neutral articles, and the terms (jargon and keywords) used in edit summaries for the languages they are currently researching. You can participate by editing the table on Phabricator, commenting on the task, or directly messaging [[m:user:Trizek (WMF)|Trizek (WMF)]].
* [[mw:Special:MyLanguage/MediaWiki Platform Team/SUL3|Single User Login]] has now been updated on all wikis to move login and account creation to a central domain. This makes user login compatible with browser restrictions on cross-domain cookies, which have prevented users of some browsers from staying logged in.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:35}} community-submitted {{PLURAL:35|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Starting on March 31st, the MediaWiki Interfaces team will begin a limited release of generated OpenAPI specs and a SwaggerUI-based sandbox experience for [[mw:Special:MyLanguage/API:REST API|MediaWiki REST APIs]]. They invite developers from a limited group of non-English Wikipedia communities (Arabic, German, French, Hebrew, Interlingua, Dutch, Chinese) to review the documentation and experiment with the sandbox in their preferred language. In addition to these specific Wikipedia projects, the sandbox and OpenAPI spec will be available on the [[testwiki:Special:RestSandbox|on the test wiki REST Sandbox special page]] for developers with English as their preferred language. During the preview period, the MediaWiki Interfaces Team also invites developers to [[mw:MediaWiki Interfaces Team/Feature Feedback/REST Sandbox|share feedback about your experience]]. The preview will last for approximately 2 weeks, after which the sandbox and OpenAPI specs will be made available across all wiki projects.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.23|MediaWiki]]

'''In depth'''
* Sometimes a small, [[gerrit:c/operations/cookbooks/+/1129184|one line code change]] can have great significance: in this case, it means that for the first time in years we're able to run all of the stack serving &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[http://maps.wikimedia.org/ maps.wikimedia.org]&lt;/bdi&gt; - a host dedicated to serving our wikis and their multi-lingual maps needs - from a single core datacenter, something we test every time we perform a [[m:Special:MyLanguage/Tech/Server switch|datacenter switchover]]. This is important because it means that in case one of our datacenters is affected by a catastrophe, we'll still be able to serve the site. This change is the result of [[phab:T216826|extensive work]] by two developers on porting the last component of the maps stack over to [[w:en:Kubernetes|kubernetes]], where we can allocate resources more efficiently than before, thus we're able to withstand more traffic in a single datacenter. This work involved a lot of complicated steps because this software, and the software libraries it uses, required many long overdue upgrades. This type of work makes the Wikimedia infrastructure more sustainable.

'''Meetings and events'''
* [[mw:Special:MyLanguage/MediaWiki Users and Developers Workshop Spring 2025|MediaWiki Users and Developers Workshop Spring 2025]] is happening in Sandusky, USA, and online, from 14–16 May 2025. The workshop will feature discussions around the usage of MediaWiki software by and within companies in different industries and will inspire and onboard new users. Registration and presentation signup is now available at the workshop's website.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:05, 1 April 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28473566 --&gt;

== Tech News: 2025-15 ==

&lt;section begin=&quot;technews-2025-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/15|Translations]] are available.

'''Updates for editors'''
* From now on, [[m:Special:MyLanguage/Interface administrators|interface admins]] and [[m:Special:MyLanguage/Central notice administrators|centralnotice admins]] are technically required to enable [[m:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]] before they can use their privileges. In the future this might be expanded to more groups with advanced user-rights. [https://phabricator.wikimedia.org/T150898]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* The Design System Team is preparing to release the next major version of Codex (v2.0.0) on April 29. Editors and developers who use CSS from Codex should see the [[mw:Codex/Release Timeline/2.0|2.0 overview documentation]], which includes guidance related to a few of the breaking changes such as &lt;code dir=ltr style=&quot;white-space: nowrap;&quot;&gt;font-size&lt;/code&gt;, &lt;code dir=ltr style=&quot;white-space: nowrap;&quot;&gt;line-height&lt;/code&gt;, and &lt;code dir=ltr style=&quot;white-space: nowrap;&quot;&gt;size-icon&lt;/code&gt;.
* The results of the [[mw:Developer Satisfaction Survey/2025|Developer Satisfaction Survey (2025)]]  are now available. Thank you to all participants. These results help the Foundation decide what to work on next and to review what they recently worked on.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.24|MediaWiki]]

'''Meetings and events'''
* The [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]] will take place in Istanbul, Turkey, between 2–4 May. Registration for attending the in-person event will close on 13 April. Before registering, please note the potential need for a [https://www.mfa.gov.tr/turkish-representations.en.mfa visa] or [https://www.mfa.gov.tr/visa-information-for-foreigners.en.mfa e-visa] to enter the country.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:52, 7 April 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28507470 --&gt;

== Tech News: 2025-16 ==

&lt;section begin=&quot;technews-2025-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/16|Translations]] are available.

'''Weekly highlight'''
* Later this week, the default thumbnail size will be increased from 220px to 250px. This changes how pages are shown in all wikis and has been requested by some communities for many years, but wasn't previously possible due to technical limitations. [https://phabricator.wikimedia.org/T355914]
* File thumbnails are now stored in discrete sizes. If a page specifies a thumbnail size that's not among the standard sizes (20, 40, 60, 120, 250, 330, 500, 960), then MediaWiki will pick the closest larger thumbnail size but will tell the browser to downscale it to the requested size. In these cases, nothing will change visually but users might load slightly larger images. If it doesn't matter which thumbnail size is used in a page, please pick one of the standard sizes to avoid the extra in-browser down-scaling step. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Images#Thumbnail_sizes][https://phabricator.wikimedia.org/T355914]

'''Updates for editors'''
* The Wikimedia Foundation are working on a system called [[m:Edge Uniques|Edge Uniques]] which will enable [[:w:en:A/B testing|A/B testing]], help protect against [[:w:en:Denial-of-service attack|Distributed denial-of-service attacks]] (DDoS attacks), and make it easier to understand how many visitors the Wikimedia sites have. This is so that they can more efficiently build tools which help readers, and make it easier for readers to find what they are looking for.
* To improve security for users, a small percentage of logins will now require that the account owner input a one-time password [[mw:Special:MyLanguage/Help:Extension:EmailAuth|emailed to their account]]. It is recommended that you [[Special:Preferences#mw-prefsection-personal-email|check]] that the email address on your account is set correctly, and that it has been confirmed, and that you have an email set for this purpose. [https://phabricator.wikimedia.org/T390662]
* &quot;Are you interested in taking a short survey to improve tools used for reviewing or reverting edits on your Wiki?&quot; This question will be [[phab:T389401|asked at 7 wikis starting next week]], on Recent Changes and Watchlist pages. The [[mw:Special:MyLanguage/Moderator Tools|Moderator Tools team]] wants to know more about activities that involve looking at new edits made to your Wikimedia project, and determining whether they adhere to your project's policies.
* On April 15, the full Wikidata graph will no longer be supported on &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[https://query.wikidata.org/ query.wikidata.org]&lt;/bdi&gt;. After this date, scholarly articles will be available through &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot; style=&quot;white-space:nowrap;&quot;&gt;[https://query-scholarly.wikidata.org/ query-scholarly.wikidata.org]&lt;/bdi&gt;, while the rest of the data hosted on Wikidata will be available through the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[https://query.wikidata.org/ query.wikidata.org]&lt;/bdi&gt; endpoint. This is part of the scheduled split of the Wikidata Graph, which was [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/September 2024 scaling update|announced in September 2024]]. More information is [[d:Wikidata:SPARQL query service/WDQS graph split|available on Wikidata]].
* The latest quarterly [[m:Special:MyLanguage/Wikimedia Apps/Newsletter/First quarter of 2025|Wikimedia Apps Newsletter]] is now available. It covers updates, experiments, and improvements made to the Wikipedia mobile apps.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* The latest quarterly [[mw:Technical Community Newsletter/2025/April|Technical Community Newsletter]] is now available. This edition includes: an invitation for tool maintainers to attend the Toolforge UI Community Feedback Session on April 15th; recent community metrics; and recent technical blog posts.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.25|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:24, 15 April 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28540654 --&gt;

== Tech News: 2025-17 ==

&lt;section begin=&quot;technews-2025-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/17|Translations]] are available.

'''Updates for editors'''
* [[f:Special:MyLanguage/Wikifunctions:Main Page|Wikifunctions]] is now integrated with [[w:dag:Solɔɣu|Dagbani Wikipedia]] since April 15. It is the first project that will be able to call [[f:Special:MyLanguage/Wikifunctions:Introduction|functions from Wikifunctions]] and integrate them in articles. A function is something that takes one or more inputs and transforms them into a desired output, such as adding up two numbers, converting miles into metres, calculating how much time has passed since an event, or declining a word into a case. Wikifunctions will allow users to do that through a simple call of [[f:Special:MyLanguage/Wikifunctions:Catalogue|a stable and global function]], rather than via a local template. [https://www.wikifunctions.org/wiki/Special:MyLanguage/Wikifunctions:Status_updates/2025-04-16]
* A new type of lint error has been created: [[Special:LintErrors/empty-heading|{{int:linter-category-empty-heading}}]] ([[mw:Special:MyLanguage/Help:Lint errors/empty-heading|documentation]]). The [[mw:Special:MyLanguage/Help:Extension:Linter|Linter extension]]'s purpose is to identify wikitext patterns that must or can be fixed in pages and provide some guidance about what the problems are with those patterns and how to fix them. [https://phabricator.wikimedia.org/T368722]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:37}} community-submitted {{PLURAL:37|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Following its publication on HuggingFace, the &quot;Structured Contents&quot; dataset, developed by Wikimedia Enterprise, is [https://enterprise.wikimedia.com/blog/kaggle-dataset/ now also available on Kaggle]. This Beta initiative is focused on making Wikimedia data more machine-readable for high-volume reusers. They are releasing this beta version in a location that open dataset communities already use, in order to seek feedback, to help improve the product for a future wider release. You can read more about the overall [https://enterprise.wikimedia.com/blog/structured-contents-snapshot-api/#open-datasets Structured Contents project], and about the [https://enterprise.wikimedia.com/blog/structured-contents-wikipedia-infobox/ first release that's freely usable].
* There is no new MediaWiki version this week.

'''Meetings and events'''
* The Editing and Machine Learning Teams invite interested volunteers to a video meeting to discuss [[mw:Special:MyLanguage/Edit check/Peacock check|Peacock check]], which is the latest [[mw:Special:MyLanguage/Edit check|Edit check]] that will detect &quot;peacock&quot; or &quot;overly-promotional&quot; or &quot;non-neutral&quot; language whilst an editor is typing. Editors who work with newcomers, or help to fix this kind of writing, or are interested in how we use artificial intelligence in our projects are encouraged to attend. The [[mw:Special:MyLanguage/Editing team/Community Conversations#Next Conversation|meeting will be on April 28, 2025]] at [https://zonestamp.toolforge.org/1745863200 18:00–19:00 UTC] and hosted on Zoom.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:00, 21 April 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28578245 --&gt;

== Tech News: 2025-18 ==

&lt;section begin=&quot;technews-2025-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/18|Translations]] are available.

'''Updates for editors'''
* Event organizers who host collaborative activities on [[m:Special:MyLanguage/CampaignEvents/Deployment status#Global Deployment Plan|multiple wikis]], including Bengali, Japanese, and Korean Wikipedias, will have access to the [[mw:Special:MyLanguage/Extension:CampaignEvents|CampaignEvents extension]] this week. Also, admins in the Wikipedia where the extension is enabled will automatically be granted the event organizer right soon. They won't have to manually grant themselves the right before they can manage events as [[phab:T386861|requested by a community]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* The release of the next major version of [[mw:Special:MyLanguage/Codex|Codex]], the design system for Wikimedia, is scheduled for 29 April 2025. Technical editors will have access to the release by the week of 5 May 2025. This update will include a number of [[mw:Special:MyLanguage/Codex/Release_Timeline/2.0#Breaking_changes|breaking changes]] and minor [[mw:Special:MyLanguage/Codex/Release_Timeline/2.0#Visual_changes|visual changes]]. Instructions on handling the breaking and visual changes are documented on [[mw:Special:MyLanguage/Codex/Release Timeline/2.0#|this page]]. Pre-release testing is reported in [[phab:T386298|T386298]], with post-release issues tracked in [[phab:T392379|T392379]] and [[phab:T392390|T392390]].
* Users of [[wikitech:Special:MyLanguage/Help:Wiki_Replicas|Wiki Replicas]] will notice that the database views of &lt;code dir=&quot;ltr&quot;&gt;ipblocks&lt;/code&gt;, &lt;code dir=&quot;ltr&quot;&gt;ipblocks_ipindex&lt;/code&gt;, and &lt;code dir=&quot;ltr&quot;&gt;ipblocks_compat&lt;/code&gt; are [[phab:T390767|now deprecated]]. Users can query the &lt;code dir=&quot;ltr&quot;&gt;[[mw:Special:MyLanguage/Manual:Block_table|block]]&lt;/code&gt; and &lt;code dir=&quot;ltr&quot;&gt;[[mw:Special:MyLanguage/Manual:Block_target_table|block_target]]&lt;/code&gt; new views that mirror the new tables in the production database instead. The deprecated views will be removed entirely from Wiki Replicas in June, 2025.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.27|MediaWiki]]

'''In depth'''
* The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/April|Language and Internationalization Newsletter]] is now available. This edition includes an overview of the improved [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;campaign=contributionsmenu&amp;to=es&amp;filter-type=automatic&amp;filter-id=previous-edits&amp;active-list=suggestions&amp;from=en#/ Content Translation Dashboard Tool], [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/April#Language Support for New and Existing Languages|support for new languages]], [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/April#Wiki Loves Ramadan Articles Made In Content Translation Mobile Workflow|highlights from the Wiki Loves Ramadan campaign]], [[m:Special:MyLanguage/Research:Languages Onboarding Experiment 2024 - Executive Summary|results from the Language Onboarding Experiment]], an analysis of topic diversity in articles, and information on upcoming community meetings and events.

'''Meetings and events'''
* The [[Special:MyLanguage/Grants:Knowledge_Sharing/Connect/Calendar|Let's Connect Learning Clinic]] will take place on [https://zonestamp.toolforge.org/1745937000 April 29 at 14:30 UTC]. This edition will focus on &quot;Understanding and Navigating Conflict in Wikimedia Projects&quot;. You can [[m:Special:MyLanguage/Event:Learning Clinic %E2%80%93 Understanding and Navigating Conflict in Wikimedia Projects (Part_1)|register now]] to attend.
* The [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]], which brings the global technical community together to connect, brainstorm, and hack existing projects, will take place from May 2 to 4th, 2025, at Istanbul, Turkey.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:31, 28 April 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28585685 --&gt;

== Tech News: 2025-19 ==

&lt;section begin=&quot;technews-2025-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/19|Translations]] are available.

'''Weekly highlight'''
* The Wikimedia Foundation has shared the latest draft update to their [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026|annual plan]] for next year (July 2025–June 2026). This includes an [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026|executive summary]] (also on [[diffblog:2025/04/25/sharing-the-wikimedia-foundations-2025-2026-draft-annual-plan/|Diff]]), details about the three main [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Goals|goals]] ([[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Product &amp; Technology OKRs|Infrastructure]], [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Goals/Volunteer Support|Volunteer Support]], and [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Goals/Effectiveness|Effectiveness]]), [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Global Trends|global trends]], and the [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Budget Overview|budget]] and [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026/Financial Model|financial model]]. Feedback and questions are welcome on the [[m:Talk:Wikimedia Foundation Annual Plan/2025-2026|talk page]] until the end of May.

'''Updates for editors'''
* For wikis that have the [[m:Special:MyLanguage/CampaignEvents/Deployment status|CampaignEvents extension enabled]], two new feature improvements have been released:
** Admins can now choose which namespaces are permitted for [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] via [[mw:Special:MyLanguage/Community Configuration|Community Configuration]] ([[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Permitted namespaces|documentation]]). The default setup is for event registration to be permitted in the Event namespace, but other namespaces (such as the project namespace or WikiProject namespace) can now be added. With this change, communities like WikiProjects can now more easily use Event Registration for their collaborative activities.
** Editors can now [[mw:Special:MyLanguage/Transclusion|transclude]] the Collaboration List on a wiki page ([[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Collaboration list/Transclusion|documentation]]). The Collaboration List is an automated list of events and WikiProjects on the wikis, accessed via {{#special:AllEvents}} ([[w:en:Special:AllEvents|example]]). Now, the Collaboration List can be added to all sorts of wiki pages, such as: a wiki mainpage, a WikiProject page, an affiliate page, an event page, or even a user page.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Developers who use the &lt;code dir=ltr&gt;moment&lt;/code&gt; library in gadgets and user scripts should revise their code to use alternatives like the &lt;code dir=ltr&gt;Intl&lt;/code&gt; library or the new &lt;code dir=ltr&gt;mediawiki.DateFormatter&lt;/code&gt; library. The &lt;code dir=ltr&gt;moment&lt;/code&gt; library has been deprecated and will begin to log messages in the developer console. You can see a global search for current uses, and [[phab:T392532|ask related questions in this Phabricator task]].
* Developers who maintain a tool that queries the Wikidata term store tables (&lt;code dir=ltr style=&quot;white-space: nowrap;&quot;&gt;wbt_*&lt;/code&gt;) need to update their code to connect to a separate database cluster. These tables are being split into a separate database cluster. Tools that query those tables via the wiki replicas must be adapted to connect to the new cluster instead. [[wikitech:News/2025 Wikidata term store database split|Documentation and related links are available]]. [https://phabricator.wikimedia.org/T390954]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.44/wmf.28|MediaWiki]]

'''In depth'''
* The latest [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|Chart Project newsletter]] is available. It includes updates on preparing to expand the deployment to additional wikis as soon as this week (starting May 6) and scaling up over the following weeks, plus exploring filtering and transforming source data.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:14, 6 May 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28665011 --&gt;

== Tech News: 2025-20 ==

&lt;section begin=&quot;technews-2025-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/20|Translations]] are available.

'''Weekly highlight'''
* The [[m:Special:MyLanguage/Wikimedia URL Shortener|&quot;Get shortened URL&quot;]] link on the sidebar now includes a [[phab:T393309|QR code]]. Wikimedia site users can now use it by scanning or downloading it to quickly share and access shared content from Wikimedia sites, conveniently.

'''Updates for editors'''
* The Wikimedia Foundation is working on a system called [[m:Edge Uniques|Edge Uniques]], which will enable [[w:en:A/B testing|A/B testing]], help protect against [[w:en:Denial-of-service attack|distributed denial-of-service attacks]] (DDoS attacks), and make it easier to understand how many visitors the Wikimedia sites have. This is to help more efficiently build tools which help readers, and make it easier for readers to find what they are looking for. Tech News has [[m:Special:MyLanguage/Tech/News/2025/16|previously written about this]]. The deployment will be gradual. Some might see the Edge Uniques cookie the week of 19 May. You can discuss this on the [[m:Talk:Edge Uniques|talk page]].
* Starting May 19, 2025, Event organisers in wikis with the [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] enabled can use [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] in the project namespace (e.g., Wikipedia namespace, Wikidata namespace). With this change, communities don't need admins to use the feature. However, wikis that don't want this change can remove and add the permitted namespaces at [[Special:CommunityConfiguration/CampaignEvents]].
* The Wikipedia project now has a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q36720|Nupe]] ([[w:nup:|&lt;code&gt;w:nup:&lt;/code&gt;]]). This is a language primarily spoken in the North Central region of Nigeria. Speakers of this language are invited to contribute to [[w:nup:Tatacin feregi|new Wikipedia]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Developers can now access pre-parsed Dutch Wikipedia, amongst others (English, German, French, Spanish, Italian, and Portuguese) through the [https://enterprise.wikimedia.com/docs/snapshot/#structured-contents-snapshot-bundle-info-beta Structured Contents snapshots (beta)]. The content includes parsed Wikipedia abstracts, descriptions, main images, infoboxes, article sections, and references.
* The &lt;code dir=&quot;ltr&quot;&gt;/page/data-parsoid&lt;/code&gt; REST API endpoint is no longer in use and will be deprecated. It is [[phab:T393557|scheduled to be turned off]] on June 7, 2025.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.1|MediaWiki]]

'''In depth'''
* The [https://wikitech.wikimedia.org/wiki/News/2025_Cloud_VPS_VXLAN_IPv6_migration IPv6 support] is a newly introduced Cloud virtual network that significantly boosts Wikimedia platforms' scalability, security, and readiness for the future. If you are a technical contributor eager to learn more, check out [https://techblog.wikimedia.org/2025/05/06/wikimedia-cloud-vps-ipv6-support/ this blog post] for an in-depth look at the journey to IPv6.

'''Meetings and events'''
* The 2nd edition of 2025 of [[m:Special:MyLanguage/Afrika Baraza|Afrika Baraza]], a virtual platform for African Wikimedians to connect, will take place on [https://zonestamp.toolforge.org/1747328400 May 15 at 17:00 UTC]. This edition will focus on discussions regarding [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2025-2026|Wikimedia Annual planning and progress]].
* The [[m:Special:MyLanguage/MENA Connect Community Call|MENA Connect Community Call]], a virtual meeting for [[w:en:Middle East and North Africa|MENA]] Wikimedians to connect, will take place on [https://zonestamp.toolforge.org/1747501200 May 17 at 17:00 UTC]. You can [[m:Event:MENA Connect (Wiki_Diwan) APP Call|register now]] to attend.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:37, 12 May 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28714188 --&gt;

== Tech News: 2025-21 ==

&lt;section begin=&quot;technews-2025-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/21|Translations]] are available.

'''Weekly highlight'''
* The Editing Team and the Machine Learning Team are working on a new check for newcomers: [[mw:Edit check/Peacock check|Peacock check]]. Using a prediction model, this check will encourage editors to improve the tone of their edits, using artificial intelligence. We invite volunteers to review the first version of the Peacock language model for the following languages: Arabic, Spanish, Portuguese, English, and Japanese. Users from these wikis interested in reviewing this model are [[mw:Edit check/Peacock check/model test|invited to sign up at MediaWiki.org]]. The deadline to sign up is on May 23, which will be the start date of the test.

'''Updates for editors'''

* From May 20, 2025, [[m:Special:MyLanguage/Oversight policy|oversighters]] and [[m:Special:MyLanguage/Meta:CheckUsers|checkusers]] will need to have their accounts secured with two-factor authentication (2FA) to be able to use their advanced rights. All users who belong to these two groups and do not have 2FA enabled have been informed. In the future, this requirement may be extended to other users with advanced rights. [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|Learn more]].
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]]  [[m:Special:MyLanguage/Community Wishlist Survey 2023/Multiblocks|Multiblocks]] will begin mass deployment by the end of the month: all non-Wikipedia projects plus Catalan Wikipedia will adopt Multiblocks in the week of May 26, while all other Wikipedias will adopt it in the week of June 2. Please [[m:Talk:Community Wishlist Survey 2023/Multiblocks|contact the team]] if you have concerns. Administrators can test the new user interface now on your own wiki by browsing to [{{fullurl:Special:Block|usecodex=1}} {{#special:Block}}?usecodex=1], and can test the full multiblocks functionality [[testwiki:Special:Block|on testwiki]]. Multiblocks is the feature that makes it possible for administrators to impose different types of blocks on the same user at the same time. See the [[mw:Special:MyLanguage/Help:Manage blocks|help page]] for more information. [https://phabricator.wikimedia.org/T377121]
* Later this week, the [[{{#special:SpecialPages}}]] listing of almost all special pages will be updated with a new design. This page has been [[phab:T219543|redesigned]] to improve the user experience in a few ways, including: The ability to search for names and aliases of the special pages, sorting, more visible marking of restricted special pages, and a more mobile-friendly look. The new version can be [https://meta.wikimedia.beta.wmflabs.org/wiki/Special:SpecialPages previewed] at Beta Cluster now, and feedback shared in the task. [https://phabricator.wikimedia.org/T219543]
* The [[mw:Special:MyLanguage/Extension:Chart|Chart extension]] is being enabled on more wikis. For a detailed list of when the extension will be enabled on your wiki, please read the [[mw:Special:MyLanguage/Extension:Chart/Project#Deployment Timeline|deployment timeline]].
* [[f:Special:MyLanguage/Wikifunctions:Main Page|Wikifunctions]] will be deployed on May 27 on five Wiktionaries: [[wikt:ha:|Hausa]], [[wikt:ig:|Igbo]], [[wikt:bn:|Bengali]], [[wikt:ml:|Malayalam]], and [[wikt:dv:|Dhivehi/Maldivian]]. This is the second batch of deployment planned for the project. After deployment, the projects will be able to call [[f:Special:MyLanguage/Wikifunctions:Introduction|functions from Wikifunctions]] and integrate them in their pages. A function is something that takes one or more inputs and transforms them into a desired output, such as adding up two numbers, converting miles into metres, calculating how much time has passed since an event, or declining a word into a case. Wikifunctions will allow users to do that through a simple call of [[f:Special:MyLanguage/Wikifunctions:Catalogue|a stable and global function]], rather than via a local template.
* Later this week, the Wikimedia Foundation will publish a hub for [[diffblog:2024/07/09/on-the-value-of-experimentation/|experiments]]. This is to showcase and get user feedback on product experiments. The experiments help the Wikimedia movement [[diffblog:2023/07/13/exploring-paths-for-the-future-of-free-knowledge-new-wikipedia-chatgpt-plugin-leveraging-rich-media-social-apps-and-other-experiments/|understand new users]], how they interact with the internet and how it could affect the Wikimedia movement. Some examples are [[m:Special:MyLanguage/Future Audiences/Generated Video|generated video]], the [[m:Special:MyLanguage/Future Audiences/Roblox game|Wikipedia Roblox speedrun game]] and [[m:Special:MyLanguage/Future Audiences/Discord bot|the Discord bot]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a bug with creating an account using the API, which has now been fixed. [https://phabricator.wikimedia.org/T390751]

'''Updates for technical contributors'''
* Gadgets and user scripts that interact with [[{{#special:Block}}]] may need to be updated to work with the new [[mw:Special:MyLanguage/Help:Manage blocks|manage blocks interface]]. Please review the [[mw:Help:Manage blocks/Developers|developer guide]] for more information. If you need help or are unable to adapt your script to the new interface, please let the team know on the [[mw:Help talk:Manage blocks/Developers|talk page]]. [https://phabricator.wikimedia.org/T377121]
* The &lt;code dir=ltr&gt;mw.title&lt;/code&gt; object allows you to get information about a specific wiki page in the [[w:en:Wikipedia:Lua|Lua]] programming language. Starting this week, a new property will be added to the object, named &lt;code dir=ltr&gt;isDisambiguationPage&lt;/code&gt;. This property allows you to check if a page is a disambiguation page, without the need to write a custom function. [https://phabricator.wikimedia.org/T71441]
* [[File:Octicons-tools.svg|15px|link=|class=skin-invert|Advanced item]] User script developers can use a [[toolforge:gitlab-content|new reverse proxy tool]] to load javascript and css from [[gitlab:|gitlab.wikimedia.org]] with &lt;code dir=ltr&gt;mw.loader.load&lt;/code&gt;. The tool's author hopes this will enable collaborative development workflows for user scripts including linting, unit tests, code generation, and code review on &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;gitlab.wikimedia.org&lt;/bdi&gt; without a separate copy-and-paste step to publish scripts to a Wikimedia wiki for integration and acceptance testing. See [[wikitech:Tool:Gitlab-content|Tool:Gitlab-content on Wikitech]] for more information.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.2|MediaWiki]]

'''Meetings and events'''
* The 12th edition of [[m:Special:MyLanguage/Wiki Workshop 2025|Wiki Workshop 2025]], a forum that brings together researchers that explore all aspects of Wikimedia projects, will be held virtually on 21-22 May. Researchers can [https://pretix.eu/wikimedia/wikiworkshop2025/ register now].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W21&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:12, 19 May 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28724712 --&gt;

== Tech News: 2025-22 ==

&lt;section begin=&quot;technews-2025-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/22|Translations]] are available.

'''Weekly highlight'''
* A community-wide discussion about a very delicate issue for the development of [[m:Special:MyLanguage/Abstract Wikipedia|Abstract Wikipedia]] is now open on Meta: where to store the abstract content that will be developed through functions from Wikifunctions and data from Wikidata. The discussion is open until June 12 at [[m:Special:MyLanguage/Abstract Wikipedia/Location of Abstract Content|Abstract Wikipedia/Location of Abstract Content]], and every opinion is welcomed. The decision will be made and communicated after the consultation period by the Foundation.

'''Updates for editors'''
* Since last week, on all wikis except [[phab:T388604|the largest 20]], people using the mobile visual editor will have [[phab:T385851|additional tools in the menu bar]], accessed using the new &lt;code&gt;+&lt;/code&gt; toolbar button. To start, the new menu will include options to add: citations, hieroglyphs, and code blocks. Deployment to the remaining wikis is [[phab:T388605|scheduled]] to happen in June.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The &lt;code dir=ltr&gt;[[mw:Special:MyLanguage/Help:Extension:ParserFunctions##ifexist|#ifexist]]&lt;/code&gt; parser function will no longer register a link to its target page. This will improve the usefulness of [[{{#special:WantedPages}}]], which will eventually only list pages that are the target of an actual red link. This change will happen gradually as the source pages are updated. [https://phabricator.wikimedia.org/T14019]
* This week, the Moderator Tools team will launch [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|a new filter to Recent Changes]], starting at Indonesian Wikipedia. This new filter highlights edits that are likely to be reverted. The goal is to help Recent Changes patrollers identify potentially problematic edits. Other wikis will benefit from this filter in the future.
* Upon clicking an empty search bar, logged-out users will see suggestions of articles for further reading. The feature will be available on both desktop and mobile. Readers of Catalan, Hebrew, and Italian Wikipedias and some sister projects will receive the change between May 21 and mid-June. Readers of other wikis will receive the change later. The goal is to encourage users to read the wikis more. [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments/Search Suggestions|Learn more]].
* Some users of the Wikipedia Android app can use a new feature for readers, [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/TrivaGame|WikiGames]], a daily trivia game based on real historical events. The release has started as an A/B test, available to 50% of users in the following languages: English, French, Portuguese, Russian, Spanish, Arabic, Chinese, and Turkish.
* The [[mw:Special:MyLanguage/Extension:Newsletter|Newsletter extension]] that is available on MediaWiki.org allows the creation of [[mw:Special:Newsletters|various newsletters]] for global users. The extension can now publish new issues as section links on an existing page, instead of requiring a new page for each issue. [https://phabricator.wikimedia.org/T393844]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* The previously deprecated &lt;code dir=ltr&gt;[[mw:Special:MyLanguage/Manual:Ipblocks table|ipblocks]]&lt;/code&gt; views in [[wikitech:Help:Wiki Replicas|Wiki Replicas]] will be removed in the beginning of June. Users are encouraged to query the new &lt;code dir=ltr&gt;[[mw:Special:MyLanguage/Manual:Block table|block]]&lt;/code&gt; and &lt;code dir=ltr&gt;[[mw:Special:MyLanguage/Manual:Block target table|block_target]]&lt;/code&gt; views instead.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.3|MediaWiki]]

'''Meetings and events'''
* [[d:Special:MyLanguage/Event:Wikidata and Sister Projects|Wikidata and Sister Projects]] is a multi-day online event that will focus on how Wikidata is integrated to Wikipedia and the other Wikimedia projects. The event runs from May 29 – June 1. You can [[d:Special:MyLanguage/Event:Wikidata and Sister Projects#Sessions|read the Program schedule]] and [[d:Special:RegisterForEvent/1291|register]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:04, 26 May 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28788673 --&gt;

== Tech News: 2025-23 ==

&lt;section begin=&quot;technews-2025-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/23|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Extension:Chart|Chart extension]] is now available on all Wikimedia wikis. Editors can use this new extension to create interactive data visualizations like bar, line, area, and pie charts. Charts are designed to replace many of the uses of the legacy [[mw:Special:MyLanguage/Extension:Graph|Graph extension]].

'''Updates for editors'''
* It is now easier to configure automatic citations for your wiki within the visual editor's [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|citation generator]]. Administrators can now set a default template by using the &lt;code dir=ltr&gt;_default&lt;/code&gt; key in the local &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Citoid-template-type-map.json]]&lt;/bdi&gt; page ([[mw:Special:Diff/6969653/7646386|example diff]]). Setting this default will also help to future-proof your existing configurations when [[phab:T347823|new item types]] are added in the future. You can still set templates for individual item types as they will be preferred to the default template. [https://phabricator.wikimedia.org/T384709]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Starting the week of June 2, bots logging in using &lt;code dir=ltr&gt;action=login&lt;/code&gt; or &lt;code dir=ltr&gt;action=clientlogin&lt;/code&gt; will fail more often. This is because of stronger protections against suspicious logins. Bots using [[mw:Special:MyLanguage/Manual:Bot passwords|bot passwords]] or using a loginless authentication method such as [[mw:Special:MyLanguage/OAuth/Owner-only consumers|OAuth]] are not affected. If your bot is not using one of those, you should update it; using &lt;code dir=ltr&gt;action=login&lt;/code&gt; without a bot password was deprecated [[listarchive:list/wikitech-l@lists.wikimedia.org/message/3EEMN7VQX5G7WMQI5K2GP5JC2336DPTD/|in 2016]]. For most bots, this only requires changing what password the bot uses. [https://phabricator.wikimedia.org/T395205]
* From this week, Wikimedia wikis will allow ES2017 features in JavaScript code for official code, gadgets, and user scripts. The most visible feature of ES2017 is &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;async&lt;/code&gt;/&lt;code&gt;await&lt;/code&gt;&lt;/bdi&gt; syntax, allowing for easier-to-read code. Until this week, the platform only allowed up to ES2016, and a few months before that, up to ES2015. [https://phabricator.wikimedia.org/T381537]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.4|MediaWiki]]

'''Meetings and events'''
* Scholarship applications to participate in the [[m:Special:MyLanguage/GLAM Wiki 2025|GLAM Wiki Conference 2025]] are now open. The conference will take place from 30 October to 1 November, in Lisbon, Portugal. GLAM contributors who lack the means to support their participation can [[m:Special:MyLanguage/GLAM Wiki 2025/Scholarships|apply here]]. Scholarship applications close on June 7th.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:55, 2 June 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28819186 --&gt;

== Tech News: 2025-24 ==

&lt;section begin=&quot;technews-2025-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/24|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Trust and Safety Product|Trust and Safety Product team]] is finalizing work needed to roll out [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] on large Wikipedias later this month. The team has worked with stewards and other users with extended rights to predict and address many use cases that may arise on larger wikis, so that community members can continue to effectively moderate and patrol temporary accounts. This will be the second of three phases of deployment – the last one will take place in September at the earliest. For more information about the recent developments on the project, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|see this update]]. If you have any comments or questions, write on the [[mw:Talk:Trust and Safety Product/Temporary Accounts|talk page]], and [[m:Event:CEE Catch up Nr. 10 (June 2025)|join a CEE Catch Up]] this Tuesday.

'''Updates for editors'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] The [[mw:Special:MyLanguage/Help:Watchlist expiry|watchlist expiry]] feature allows editors to watch pages for a limited period of time. After that period, the page is automatically removed from your watchlist. Starting this week, you can set a preference for the default period of time to watch pages. The [[Special:Preferences#mw-prefsection-watchlist-pageswatchlist|preferences]] also allow you to set different default watch periods for editing existing pages, pages you create, and when using rollback. [https://phabricator.wikimedia.org/T265716]
[[File:Talk pages default look (April 2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* The appearance of talk pages will change at almost all Wikipedias ([[m:Special:MyLanguage/Tech/News/2024/19|some]] have already received this design change, [[phab:T379264|a few]] will get these changes later). You can read details about the changes [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. It is possible to opt out of these changes [[Special:Preferences#mw-prefsection-editing-discussion|in user preferences]] (&quot;{{int:discussiontools-preference-visualenhancements}}&quot;). [https://phabricator.wikimedia.org/T319146][https://phabricator.wikimedia.org/T392121]
* Users with specific extended rights (including administrators, bureaucrats, checkusers, oversighters, and stewards) can now have IP addresses of all temporary accounts [[phab:T358853|revealed automatically]] during time-limited periods where they need to combat high-speed account-hopping vandalism. This feature was requested by stewards. [https://phabricator.wikimedia.org/T386492]
* This week, the Moderator Tools and Machine Learning teams will continue the rollout of [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|a new filter to Recent Changes]], releasing it to several more Wikipedias. This filter utilizes the Revert Risk model, which was created by the Research team, to highlight edits that are likely to be reverted and help Recent Changes patrollers identify potentially problematic contributions. The feature will be rolled out to the following Wikipedias: {{int:project-localized-name-afwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-simplewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}. The rollout will continue in the coming weeks to include [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|the rest of the Wikipedias in this project]]. [https://phabricator.wikimedia.org/T391964]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* AbuseFilter editors active on Meta-Wiki and large Wikipedias are kindly asked to update AbuseFilter to make it compatible with temporary accounts. A link to the instructions and the private lists of filters needing verification are [[phab:T369611|available on Phabricator]].
* Lua modules now have access to the name of a page's associated thumbnail image, and on [https://gerrit.wikimedia.org/g/operations/mediawiki-config/+/2e4ab14aa15bb95568f9c07dd777065901eb2126/wmf-config/InitialiseSettings.php#10849 some wikis] to the WikiProject assessment information. This is possible using two new properties on [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#added-by-extensions|mw.title objects]], named &lt;code dir=ltr&gt;pageImage&lt;/code&gt; and &lt;code dir=ltr&gt;pageAssessments&lt;/code&gt;. [https://phabricator.wikimedia.org/T131911][https://phabricator.wikimedia.org/T380122]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.5|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:16, 10 June 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28846858 --&gt;

== Tech News: 2025-25 ==

&lt;section begin=&quot;technews-2025-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/25|Translations]] are available.

'''Updates for editors'''
* You can [https://wikimediafoundation.limesurvey.net/359761?lang=en nominate your favorite tools] for the sixth edition of the [[m:Special:MyLanguage/Coolest Tool Award|Coolest Tool Award]]. Nominations are anonymous and will be open until June 25. You can re-use the survey to nominate multiple tools.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:33}} community-submitted {{PLURAL:33|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.6|MediaWiki]]

'''In depth'''
* Foundation staff and technical volunteers use Wikimedia APIs to build the tools, applications, features, and integrations that enhance user experiences. Over the coming years, the MediaWiki Interfaces team will be investing in Wikimedia web (HTTP) APIs to better serve technical volunteer needs and protect Wikimedia infrastructure from potential abuse. You can [https://techblog.wikimedia.org/2025/06/12/apis-as-a-product-investing-in-the-current-and-next-generation-of-technical-contributors/ read more about their plans to evolve the APIs in this Techblog post].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:38, 16 June 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28870688 --&gt;

== Tech News: 2025-26 ==

&lt;section begin=&quot;technews-2025-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/26|Translations]] are available.

'''Weekly highlight'''
* This week, the Moderator Tools and Machine Learning teams will continue the rollout of [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|a new filter to Recent Changes]], releasing it to the third and last batch of Wikipedias. This filter utilizes the Revert Risk model, which was created by the Research team, to highlight edits that are likely to be reverted and help Recent Changes patrollers identify potentially problematic contributions. The feature will be rolled out to the following Wikipedias: {{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-swwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tlwiki/en}}. The rollout will continue in the coming weeks to include [[mw:Special:MyLanguage/2025 RecentChanges Language Agnostic Revert Risk Filtering|the rest of the Wikipedias in this project]]. [https://phabricator.wikimedia.org/T391964]

'''Updates for editors'''
* Last week, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] were rolled out on Czech, Korean, and Turkish Wikipedias. This and next week, deployments on larger Wikipedias will follow. [[mw:Talk:Trust and Safety Product/Temporary Accounts|Share your thoughts]] about the project. [https://phabricator.wikimedia.org/T340001]
* Later this week, the Editing team will release [[mw:Special:MyLanguage/Help:Edit check#Multi check|Multi Check]] to all Wikipedias (except English Wikipedia). This feature shows multiple [[mw:Special:MyLanguage/Help:Edit check#Reference check|Reference checks]] within the editing experience. This encourages users to add citations when they add multiple new paragraphs to a Wikipedia article. This feature was previously available as an A/B test. [https://analytics.wikimedia.org/published/reports/editing/multi_check_ab_test_report_final.html#summary-of-results The test shows] that users who are shown multiple checks are 1.3 times more likely to add a reference to their edit, and their edit is less likely to be reverted (-34.7%). [https://phabricator.wikimedia.org/T395519]
* A few pages need to be renamed due to software updates and to match more recent Unicode standards. All of these changes are related to title-casing changes. Approximately 71 pages and 3 files will be renamed, across 15 wikis; the complete list is in [[phab:T396903|the task]]. The developers will rename these pages next week, and they will fix redirects and embedded file links a few minutes later via a system settings update.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that had caused pages to scroll upwards when text near the top was selected. [https://phabricator.wikimedia.org/T364023]

'''Updates for technical contributors'''
* Editors can now use Lua modules to filter and transform tabular data for use with [[mw:Special:MyLanguage/Extension:Chart|Extension:Chart]]. This can be used for things like selecting a subset of rows or columns from the source data, converting between units, statistical processing, and many other useful transformations. [[mw:Special:MyLanguage/Extension:Chart/Transforms|Information on how to use transforms is available]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:Chart/Project/Updates]
* The &lt;code dir=ltr&gt;all_links&lt;/code&gt; variable in [[Special:AbuseFilter|AbuseFilter]] is now renamed to &lt;code dir=ltr&gt;new_links&lt;/code&gt; for consistency with other variables. Old usages will still continue to work. [https://phabricator.wikimedia.org/T391811]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.7|MediaWiki]]

'''In depth'''
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/34|Growth newsletter]] is available. It includes: the recent updates for the &quot;Add a Link&quot; Task, two new Newcomer Engagement Features, and updates to Community Configuration.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:21, 23 June 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28870688 --&gt;

== Tech News: 2025-27 ==

&lt;section begin=&quot;technews-2025-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/27|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] has been enabled on all Wikipedias. The extension makes it easier to organize and participate in collaborative activities, like edit-a-thons and WikiProjects, on the wikis. The extension has three features: [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], [[m:Special:MyLanguage/CampaignEvents/Collaboration list|Collaboration List]], and [[m:Campaigns/Foundation Product Team/Invitation list|Invitation List]]. To request the extension for your wiki, visit the [[m:Special:MyLanguage/CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|Deployment information page]].

'''Updates for editors'''
* AbuseFilter maintainers can now [[mw:Special:MyLanguage/Extension:IPReputation/AbuseFilter variables|match against IP reputation data]] in [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilters]]. IP reputation data is information about the proxies and VPNs associated with the user's IP address. This data is not shown publicly and is not generated for actions performed by registered accounts. [https://phabricator.wikimedia.org/T354599]
* Hidden content that is within [[mw:Special:MyLanguage/Manual:Collapsible elements|collapsible parts of wikipages]] will now be revealed when someone searches the page using the web browser's &quot;Find in page&quot; function (Ctrl+F or ⌘F) in supporting browsers. [https://phabricator.wikimedia.org/T327893][https://developer.mozilla.org/en-US/docs/Web/HTML/Reference/Global_attributes/hidden#browser_compatibility]
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new feature, called [[mw:Special:MyLanguage/Help:TemplateData/Template discovery|Favourite Templates]], will be deployed later this week on all projects (except English Wikipedia, which will receive the feature next week), following a piloting phase on Polish and Arabic Wikipedia, and Italian and English Wikisource. The feature will provide a better way for new and experienced contributors to recall and discover templates via the template dialog, by allowing users to put templates on a special &quot;favourite list&quot;. The feature works with both the visual editor and the wikitext editor. The feature is a [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|community wishlist focus area]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that had caused some Notifications to be sent multiple times. [https://phabricator.wikimedia.org/T397103]

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.8|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:40, 30 June 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28917415 --&gt;

== Tech News: 2025-28 ==

&lt;section begin=&quot;technews-2025-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/28|Translations]] are available.

'''Weekly highlight'''
* [[mw:Special:MyLanguage/Help:Temporary accounts|Temporary accounts]] have been rolled out on 18 large and medium-sized Wikipedias, including German, Japanese, French, and Chinese. Now, about 1/3 of all logged-out activity across wikis is coming from temporary accounts. Users involved in patrolling may be interested in two new documentation pages: [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Access to IP|Access to IP]], explaining everything related to access to temporary account IP addresses, and [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Repository|Repository]] with a list of new gadgets and user scripts.

'''Updates for editors'''
* Anyone can play an experimental new game, [[mw:Special:MyLanguage/New Engagement Experiments/WikiRun|WikiRun]], that lets you race through Wikipedia by clicking from one article to another, aiming to reach a target page in as few steps and in as little time as possible. The project's goal is to explore new ways of engaging readers. [https://wikirun-game.toolforge.org/ Try playing the game] and let the team know what you think [[mw:Talk:New Engagement Experiments/WikiRun|on the talk page]].
* Users of the Wikipedia Android app in some languages can now play the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/TrivaGame|trivia game]]. ''Which came first?'' is a simple history game where you guess which of two events happened earlier on today's date. It was previously available as an A/B test. It is now available to all users in English, German, French, Spanish, Portuguese, Russian, Arabic, Turkish, and Chinese. The goal of the feature is to help engage with new generations of readers. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/News/2025/22]
* Users of the iOS Wikipedia App in some languages may see a new tabbed browsing feature that enables you to open multiple tabs while reading. This feature makes it easier to explore related topics and switch between articles. The A/B test is currently running in Arabic, English, and Japanese in selected regions. More details are available on the [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Tabbed Browsing (Tabs)|Tabbed Browsing project page]].
* Bureaucrats on Wikimedia wikis can now use [[{{#special:VerifyOATHForUser}}]] to check if users have enabled [[mw:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]]. [https://phabricator.wikimedia.org/T265726]
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new feature related to [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template Recall and Discovery]] will be deployed later this week to all Wikimedia projects: a [[mw:Special:MyLanguage/Help:TemplateData/Template discovery#Template categories|template category browser]] will be introduced to assist users in finding templates to put in their “favourite” list. The browser will allow users to browse a list of templates which have been organised into a given category tree. The feature has been requested by the community [[m:Special:MyLanguage/Community Wishlist/Wishes/Select templates by categories|through the Community Wishlist]].
* It is now possible to access watchlist preferences from the watchlist page. Also the redundant button to edit the watchlist has been removed. [https://www.mediawiki.org/wiki/Moderator_Tools/Watchlist]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* As part of [[mw:MediaWiki_1.44|MediaWiki 1.44]] there is now a unified built-in Notifications system that makes it easier for developers to send, manage, and customize notifications. Check out the updated documentation at [[mw:Manual:Notifications|Manual:Notifications]], information about migration in [[phab:T388663|T388663]] and details on deprecated hooks in [[phab:T389624|T389624]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.9|MediaWiki]]

'''Meetings and events'''
* [[d:Special:MyLanguage/Event:WikidataCon 2025|WikidataCon 2025]], the conference dedicated to Wikidata is now open for [https://pretalx.com/wikidatacon-2025/cfp session proposals] and for [[d:Special:RegisterForEvent/1340|registration]]. This year's event will be held online from October 31 – November 02 and will explore on the theme of &quot;Connecting People through Linked Open Data&quot;.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:05, 8 July 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28930584 --&gt;

== Tech News: 2025-29 ==

&lt;section begin=&quot;technews-2025-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/29|Translations]] are available.

'''Updates for editors'''
* [[mw:Special:MyLanguage/Help:TemplateData/Template discovery#Featured templates|Featured templates]], a new feature related to [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template Recall and Discovery]] will be deployed this week to all Wikimedia projects: With this feature, editors will be able to quickly access a list of templates that are likely to be useful. These templates will be displayed in a list, under the &quot;featured&quot; tab of the template discovery interface. Administrators can define the list via the Community Configuration interface. The feature fulfills a request by the community [[m:Special:MyLanguage/Community Wishlist/Wishes/Easy access Templates|through the Community Wishlist]]. [https://phabricator.wikimedia.org/T367428][https://phabricator.wikimedia.org/T392896]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the request to add Malayalam fonts in the [[oldWikisource:Special:MyLanguage/Wikisource:WS Export|Wikisource Book Export Tool]] was resolved and now, the rendering of Malayalam letters in exported Wikisource books are accurate. [https://phabricator.wikimedia.org/T374457]

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.10|MediaWiki]]

'''In depth'''
* Developers, designers, and all Wikimedians are invited to [https://phabricator.wikimedia.org/project/board/7953/ submit a project idea] for the Wikimania Hackathon 2025. Read [https://diff.wikimedia.org/2025/06/30/call-for-projects-wikimania-hackathon-2025-is-coming-to-nairobi/ this Diff blog post] for more details.

'''Meetings and events'''
* [[m:WikiIndaba conference 2025|WikiIndaba 2025]] scholarship application and program submission is open until 23:59 GMT on July 20. WikiIndaba is a regional conference for African Wikimedians both on the continent and in the diaspora to unite and grow together. Submit [https://docs.google.com/forms/d/e/1FAIpQLSdJTv68R1OPASXXDfpIl8EWiMLTM-TDwh6_5gNVvFuWccFZ2Q/viewform your scholarship application] and [https://ee.kobotoolbox.org/x/BI3omIfH program proposal] now!
* [https://br.wikimedia.org/wiki/WikiCon_Brasil_2025 WikiCon Brasil 2025] will take place on July 19-20 in Salvador, Bahia, Brazil. The Brazilian community members are encouraged to register and attend!

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:09, 14 July 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=28980963 --&gt;

== Tech News: 2025-30 ==

&lt;section begin=&quot;technews-2025-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/30|Translations]] are available.

'''Updates for editors'''
* The Translation Suggestions feature in the [[mw:Special:MyLanguage/Content translation|Content Translation tool]] now has another level of article filters added to the &quot;[https://en.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;filter-type=automatic&amp;filter-id=previous-edits&amp;active-list=suggestions&amp;from=en&amp;to=fi#/ ... More]&quot; category. Translators who use the Suggestions feature can now select and receive article suggestions that are customized to geographical locations of their interest using the new &quot;{{int:Cx-sx-suggestions-filters-tab-regions}}&quot; filter. [https://phabricator.wikimedia.org/T113257]
* Administrators can now limit &quot;Add a Link&quot; to newcomers. The [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|&quot;Add a Link&quot;]] Structured Task [[mw:Special:MyLanguage/Growth/Constructive activation experimentation#Enwiki A/B test &amp; &quot;Add a Link&quot; Improvements (Wiki Experiences 1.2.11 &amp; 1.2.16)|helps new account holders start editing]], but some communities have requested the ability to restrict it to its intended audience: newcomers. Administrators can configure this setting within the [[Special:CommunityConfiguration/GrowthSuggestedEdits|Community Configuration]] feature.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* For AbuseFilter editors on [[phab:T392144|some wikis]], it is now possible to filter edits based on the RevertRisk score of the edit being attempted. It is only populated if the action being evaluated is an edit. For more information, please see the [[mw:Special:MyLanguage/Extension:ORES/AbuseFilter variables#What variables are available for use|ORES/AbuseFilter variables]] documentation.
* The [[mw:Special:MyLanguage/Beta Cluster|Beta Cluster]] wikis have [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/YDABPV75LADRQCXMJAFWUP256N4EQ25B/|been moved]] from &lt;code dir=ltr&gt;beta.wmflabs.org&lt;/code&gt; to &lt;code dir=ltr&gt;beta.wmcloud.org&lt;/code&gt;. Users may need to update URLs in any tools, or in their password managers. Any related issues can be [[phab:T289318|reported in the task]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.11|MediaWiki]]

'''Meetings and events'''
* [[m:Special:MyLanguage/WikiCite 2025|WikiCite 2025]] will take place from 29–31 August, both online and in-person in Bern, Switzerland. The event's goals are to reconnect communities, institutions, and individuals working with open citations, bibliographic data, and the Wikidata/Wikibase ecosystem. Registration is open and the call for proposals will be announced soon. [https://lists.wikimedia.org/hyperkitty/list/wikidata@lists.wikimedia.org/message/KQZUG3ETKLBWPBYSB2YAWZIRPWHS24TG/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W30&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:42, 21 July 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29005283 --&gt;

== Tech News: 2025-31 ==

&lt;section begin=&quot;technews-2025-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/31|Translations]] are available.

'''Weekly highlight'''
* The Community Tech team will be focusing on wishes related to Watchlists and Recent Changes pages, over the next few months. They are looking for feedback. Please [[m:Special:MyLanguage/Community Wishlist/Updates#July 24, 2025: Watchlists and Recent Changes pages|read the latest update]], and if you have ideas, please [[m:Special:MyLanguage/Community Wishlist|submit a wish]] on the topic.

'''Updates for editors'''
* The Wikimedia Commons community has decided to block [[:mw:Special:MyLanguage/Upload dialog|cross-wiki uploads]] to Wikimedia Commons, for all users without autoconfirmed rights on that wiki, starting on August 16. This is because of [[:c:Commons:Cross-wiki media upload tool/History|widespread problems]] related to files that are uploaded by newcomers. Users who are affected by this will get an error message with a link to the less restrictive UploadWizard on Commons. Please help translating the [[:c:Special:MyLanguage/MediaWiki:Abusefilter-disallowed-cross-wiki-upload|message]] or give feedback on the message text. Please also update your local help pages to explain this restriction. [https://phabricator.wikimedia.org/T370598]
* On wikis with temporary accounts enabled and Meta-Wiki, administrators may now set up a footer for the Special:Contributions pages of temporary accounts, similar to those which can be shown on IP and user-account pages. They may do it by creating the page named &lt;code dir=ltr&gt;MediaWiki:Sp-contributions-footer-temp&lt;/code&gt;. [https://phabricator.wikimedia.org/T398347]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.12|MediaWiki]]

'''Meetings and events'''
* [[wmania:Special:MyLanguage/2025:Wikimania|Wikimania 2025]] will run from August 6–9. The [https://wikimedia.eventyay.com/talk/wikimania2025/schedule/ program is available] for you to plan which sessions you want to attend. Most sessions will be live-streamed, with exceptions for those that show the &quot;no camera&quot; icon. If you are joining online to watch live-streams and use the interactive features, please [[wmania:Special:MyLanguage/2025:Registration|register]] for a free virtual ticket. For example, you may be interested in technical sessions such as:
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/KFEFVG/ Temporary Accounts: Enhancing privacy for our unregistered editors]
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/TVCVAB/ Building a Sustainable Future for Wikimedia Contributors]
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/WTRQCJ/ A dozen visions for wikitext!]
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/8YKKP9/ Coordinate Across Stakeholders with the Product and Technology Advisory Council]
* The [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2025|MediaWiki Users and Developers Conference, Fall 2025]] will be held 28–30 October 2025 in Hanover, Germany. This event is organized by and for the third-party MediaWiki community. You can propose sessions and register to attend.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:26, 29 July 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29051727 --&gt;

== Tech News: 2025-32 ==

&lt;section begin=&quot;technews-2025-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/32|Translations]] are available.

'''Updates for editors'''
* Editors can now enable the [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/User Info|User Info card]]. This feature adds an icon next to usernames on history pages and similar user-contribution log pages. When you tap or click on the icon, it displays data related to that user account such as the number of edits, reverted edits, blocks, and more. It's part of a broader project to make it easier for moderators to evaluate account trustworthiness. The feature can be enabled in [[testwiki:Special:GlobalPreferences#mw-prefsection-rendering|your global preferences]], and later this week it will be available in local preferences. [https://phabricator.wikimedia.org/T386439]
* Everybody is invited to share comments on [[m:Special:MyLanguage/CampaignEvents/Collaborative contributions|Collaborative Contributions]], a project recently launched by the [[m:Special:MyLanguage/Connection Team|Connection team]]. The project aims to create a new way to display the impact of collaborative editing activities (such as edit-a-thons, backlog drives, and WikiProjects) on the wikis. Post your comments on the [[m:Talk:CampaignEvents/Collaborative contributions|project talk page]]. [https://phabricator.wikimedia.org/T378035]
* Administrators can now define the default block duration for temporary accounts. To do that, they need to create a page named &lt;code dir=ltr&gt;MediaWiki:Ipb-default-expiry-temporary-account&lt;/code&gt; and use a value defined in &lt;code dir=ltr&gt;MediaWiki:Ipboptions&lt;/code&gt;. This allows administrators to easily block temporary accounts for 90 days, which is functionally equivalent to an indefinite block. The advantage of this solution is that it does not clutter Special:BlockList. [[mw:Special:MyLanguage/Manual:Block and unblock#Default block duration options|More documentation]] is available. [https://phabricator.wikimedia.org/T398626]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Gadgets can now include &lt;code dir=ltr&gt;.vue&lt;/code&gt; files. This makes it easier to develop modern user interfaces using [[mw:Vue.js|Vue.js]], in particular using [[mw:Special:MyLanguage/Codex|Codex]], the official design system of Wikimedia. [[wmdoc:codex/latest/icons/overview.html|Codex icons]] can be loaded through the gadget definition. [[mw:Special:MyLanguage/Extension:Gadgets#Pages|The documentation]] has examples. For user scripts that use Vue.js, an [[mw:API:CodexIcons|API module]] now exists to load Codex icons. [https://phabricator.wikimedia.org/T340460][https://phabricator.wikimedia.org/T311099]
* Module developers can now use a [[mw:Help:Extension:Translate/Message Bundles/Lua reference|Lua interface]] to simplify the preparation of Lua modules for translation on Meta-Wiki. This improvement makes it easier for translators to find and edit module strings without dealing with raw Lua code. It helps prevent mistakes that could break the module during translation. Module developers and translators are invited to [[commons:File:Translatable modules video demo July 2025.webm|watch the demo video]], read more about [[mw:Special:MyLanguage/Translatable modules|translatable modules]] to understand how it works, refer to Meta-Wiki's [[m:Module:User Wikimedia project|Module:User Wikimedia project]] for example usage, and [[mw:Talk:Translatable modules|share their feedback]] on how well it addresses the challenges in their workflow. The interface still has some performance issues, so it should not be used in widely used modules yet. [https://phabricator.wikimedia.org/T359918]
* Developers of external tools that connect to Wikimedia pages must set a user-agent that complies with [[foundation:Special:MyLanguage/Policy:Wikimedia Foundation User-Agent Policy|the user-agent policy]]. This policy will start to be more strongly enforced in August because of external crawlers that are [[diffblog:2025/04/01/how-crawlers-impact-the-operations-of-the-wikimedia-projects/|overusing]] Wikimedia's resources. Tools that are hosted on Wikimedia's Toolforge or Cloud VPS will not be affected by this for now, but should still set a user-agent. [[phab:T400119|More technical details are available]], and related questions are welcome in that task.
* Parsoid Read Views is going to be rolling out to some smaller Wikipedias over the next few weeks, following the successful transition of Wikivoyages and Wiktionaries to Parsoid Read Views. For more information, see the [[mw:Special:MyLanguage/Parsoid/Parser Unification|Parsoid/Parser Unification]] project page. [https://phabricator.wikimedia.org/project/profile/7694/]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.13|MediaWiki]]

'''Meetings and events'''
* [[wmania:Special:MyLanguage/2025:Wikimania|Wikimania 2025]] will run from August 6–9. The [https://wikimedia.eventyay.com/talk/wikimania2025/schedule/ program is available] for you to plan which sessions you want to attend. Most sessions will be live-streamed, with exceptions for those that show the &quot;no camera&quot; icon. If you are joining online to watch live-streams and use the interactive features, please [[wmania:Special:MyLanguage/2025:Registration|register]] for a free virtual ticket. For example, you may be interested in technical sessions such as:
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/GEH9DH/ Wikimedia’s knowledge infrastructure in a changing internet: Establishing sustainable pathways for content reuse]
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/7ELN9Q/ Wikifunctions is coming soon to a wiki near you!]
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/ZMGVJV/ Shaping the Future of Wikipedia’s Reader Experience]
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/KCKTFZ/ Making Wikipedia More Readable: What Comes Next]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 03:40, 5 August 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29083927 --&gt;

== Tech News: 2025-33 ==

&lt;section begin=&quot;technews-2025-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/33|Translations]] are available.

'''Updates for editors'''
* The WikiEditor toolbar now includes [[mw:Special:MyLanguage/Help:Extension:WikiEditor#Keyboard shortcuts|its keyboard shortcuts]] in the tooltips for its buttons. This will help to improve the discoverability of this feature. [https://phabricator.wikimedia.org/T400583]
* The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] published a set of [[m:Special:MyLanguage/Product and Technology Advisory Council/August 2025 draft PTAC proposals for feedback|proposed experiments]] the Wikimedia Foundation can try to improve communication with community. Feedback on the proposals are welcomed until August 22 on [[m:Talk:Product and Technology Advisory Council/August 2025 draft PTAC proposals for feedback|this talk page]].
* The search bar on the Minerva skin (mobile) has been updated to use the same type-ahead search component that is used on the Vector 2022 skin. There are no changes in search functionality but there are minor visual changes. Specifically, the close-search button has been changed from an &quot;X&quot; to a back arrow. This helps to distinguish it from the other &quot;X&quot; button that is used to clear any text. [https://phabricator.wikimedia.org/T393944]
* Editors on some wikis will see a new toggle for &quot;Group results by page&quot; on watchlist, related changes, and recent changes pages. This is [[mw:Special:MyLanguage/Moderator Tools/Watchlist/Experiment|an A/B experiment]] that is planned to start on August 11, and will run for 3–6 weeks on the Bengali, Chinese, Czech, French, Greek, Portuguese, and Urdu Wikipedias. The experiment will examine how making this feature more discoverable might affect editors' ability to find the edits they are looking for. [https://phabricator.wikimedia.org/T396789]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* The multiwiki datasets of [[:wikt:en:Module:Unicode data|Unicode data]] have been moved to [[c:Category:Unicode Module Datasets|Category:Unicode Module Datasets]] on Wikimedia Commons, to follow the idea of &quot;One common data source, multiple local wikis&quot;. Most wikis have been updated to use the Commons version. You can ask questions at [[c:Category talk:Unicode Module Datasets|the talkpage]]. [https://en.wiktionary.org/wiki/Module_talk:Unicode_data#Data_from_commons]
* Lua code can add warnings when something is wrong, by using the &lt;code dir=ltr&gt;mw.addWarning()&lt;/code&gt; function. It is now possible to add more than one warning, instead of new warnings replacing old ones. If you maintain a Lua module that used warnings, you should check it still works as expected. [https://phabricator.wikimedia.org/T398390]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.14|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:29, 11 August 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29106516 --&gt;

== Tech News: 2025-34 ==

&lt;section begin=&quot;technews-2025-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/34|Translations]] are available.

'''Updates for editors'''
* Later this week, people who are logged-in and have the &quot;[[mw:Special:MyLanguage/Talk pages project/Feature summary|Discussion tools]]&quot; [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] enabled will gain the ability to &quot;Thank&quot; individual comments directly from talk pages, rather than needing to navigate to page history. [[mw:Special:MyLanguage/Talk pages project/Feature summary#Comment actions|Learn more about this feature]]. [https://phabricator.wikimedia.org/T400849]
* An A/B test comparing two versions of the desktop donate link launched on testwiki on 12 August and on English Wikipedia 14 August for 0.1% of logged out users on the desktop site. The experiment will run for three weeks, ending on 12 September. [https://phabricator.wikimedia.org/T395716]
* An A/A test to measure the baseline for reader retention was launched 12 August using [[wikitech:Experimentation Lab|Experimentation Lab]]. This measures the percentage of users who revisit a wiki after their initial visit over a 14-day period. No visual changes are expected. The experiment will run through 31 August. [https://phabricator.wikimedia.org/T399227]
* Five new wikis have been created:
** a {{int:project-localized-name-group-wikisource/en}} in [[d:Q34057|Tagalog]] ([[s:tl:|&lt;code&gt;s:tl:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T388639]
** a {{int:project-localized-name-group-wikisource/en}} in [[d:Q36213|Madurese]] ([[s:mad:|&lt;code&gt;s:mad:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T391747]
** a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q3450749|Rakhine]] ([[w:rki:|&lt;code&gt;w:rki:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T392490]
** a {{int:project-localized-name-group-wikibooks/en}} in [[d:Q13324|Minangkabau]] ([[b:min:|&lt;code&gt;b:min:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T395452]
** a {{int:project-localized-name-group-wiktionary/en}} in [[d:Q7598268|Standard Moroccan Amazigh]] ([[wikt:zgh:|&lt;code&gt;wikt:zgh:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T399684]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:46}} community-submitted {{PLURAL:46|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.15|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W34&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:38, 19 August 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29127690 --&gt;

== Tech News: 2025-35 ==

&lt;section begin=&quot;technews-2025-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/35|Translations]] are available.

'''Updates for editors'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Template authors can now use additional CSS properties, since the CSS sanitizer used by [[mw:Special:MyLanguage/Help:TemplateStyles|TemplateStyles]] was updated. For example: &lt;code&gt;width: fit-content&lt;/code&gt;; &lt;code&gt;ruby-align&lt;/code&gt;; relative units such as &lt;code&gt;lh&lt;/code&gt;; and custom strings in &lt;code&gt;list-style-type&lt;/code&gt;. These improvements are a [[m:Special:MyLanguage/Community Wishlist/Wishes/Allow use of modern CSS in templates by updating the TemplateStyles CSS sanitizer|Community Wishlist wish]]. [https://phabricator.wikimedia.org/T271958][https://phabricator.wikimedia.org/T277755][https://phabricator.wikimedia.org/T293633][https://phabricator.wikimedia.org/T295088][https://phabricator.wikimedia.org/T326906][https://phabricator.wikimedia.org/T340057][https://phabricator.wikimedia.org/T360725][https://phabricator.wikimedia.org/T371809][https://phabricator.wikimedia.org/T375344][https://phabricator.wikimedia.org/T394619]
* On large wikis, the default time period to display edits from, within the Special:RecentChanges page, has been changed from 7 days to 1 day. This is part of a performance improvement project. This should have no user-facing impact due to the quantity of edits on these wikis. [https://phabricator.wikimedia.org/T399455]
* Administrators can now access the [[{{#special:BlockedExternalDomains}}]] page from the [[{{#special:CommunityConfiguration}}]] list page. This makes it easier to find. [https://phabricator.wikimedia.org/T393240]
* Wikimedia Commons videos were not shown in the Videos tab in Google Search. The problem was investigated and reported to Google who have now fixed the issue. [https://phabricator.wikimedia.org/T396168][https://meta.wikimedia.org/wiki/Community_Wishlist/Wishes/Do_something_about_Google_%26_DuckDuckGo_search_not_indexing_media_files_and_categories_on_Commons] 
* One new wiki has been created: a {{int:project-localized-name-group-wiktionary/en}} in [[d:Q33014|Betawi]] ([[wikt:bew:|&lt;code&gt;wikt:bew:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T402130]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:39}} community-submitted {{PLURAL:39|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Two fields of the [[mw:Special:MyLanguage/Manual:Recentchanges table|recentchanges database table]] are being removed. &lt;code&gt;rc_new&lt;/code&gt; and &lt;code&gt;rc_type&lt;/code&gt; are being removed in favor of &lt;code&gt;rc_source&lt;/code&gt;. Queries to these older fields will start to fail starting this week and developers should use &lt;code&gt;rc_source&lt;/code&gt; instead. These older fields were deprecated over 10 years ago and should not be in use. This is part of work to improve the performance and stability of queries to the recentchanges table. [https://phabricator.wikimedia.org/T400696] 
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.16|MediaWiki]]

'''In depth'''
* The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2025/July|Language and Internationalization Newsletter]] is now available. This edition includes: support for new languages in MediaWiki and translatewiki; the start of the Language Onboarding and Development project to help support the growth of new and small wikis; updates on research projects; and more.

'''Meetings and events'''
* The next [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Community meetings#29 August 2025|Language Community Meeting]] is happening soon, August 29th at [https://zonestamp.toolforge.org/1756479600 15:00 UTC]. This week's meeting will cover: the Avro keyboard developers from Wikimedia Bangladesh, who were recently awarded a national award for their contributions to this keyboard; and other topics. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:12, 26 August 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29175124 --&gt;

== Tech News: 2025-36 ==

&lt;section begin=&quot;technews-2025-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/36|Translations]] are available.

'''Weekly highlight'''
* The Editing team wants to compile a list of templates, jargon terms, and policies used in edit summaries when a copyright violation is removed. This will help them identify the number of edits reverted due to copyright issues. We invite community members from the following Wikis to list these terms in [[Phab:T402601|T402601]], or to share their list with [[User:Trizek (WMF)|Trizek_(WMF)]]: {{int:project-localized-name-arwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-enwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-plwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-viwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zhwiki/en}}. This project is open until September 9th 2025.

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] has been enabled for all Wikisources. The extension makes it easier to organize and participate in collaborative activities, like edit-a-thons and WikiProjects, on the wikis. The extension has three features: [[m:Special:MyLanguage/Event Center/Registration|Event Registration]], [[m:Special:MyLanguage/CampaignEvents/Collaboration list|Collaboration List]], and [[m:Special:MyLanguage/Connection Team/Invitation list|Invitation List]]. To request the extension for your wiki, visit the Deployment information page. [https://meta.wikimedia.org/wiki/CampaignEvents/Deployment_status#How_to_Request_the_CampaignEvents_Extension_for_your_wiki]
* The lists in the footer of the editing interface, such as &quot;Templates used on this page,&quot; will now be organized into columns when there is enough space. This enhancement minimizes scrolling when editing lengthy articles on Wikipedia. [https://phabricator.wikimedia.org/T401066]
* On September 3rd, 2025 we will increase the sampling percentages of our [[mw:Special:MyLanguage/Moderator Tools/Watchlist/Experiment#Scope of the experiment|group by toggle experiment]] of the &lt;code&gt;Special:RecentChanges&lt;/code&gt;, &lt;code&gt;Special:Watchlist&lt;/code&gt;, and &lt;code&gt;Special:RelatedChanges&lt;/code&gt; pages on the Chinese, French, and Portuguese Wikipedias to 100 percent, allowing more editors to be part of this experiment. This adjustment is intended to ensure we have sufficient data to make informed decisions when evaluating the experiment results. [https://phabricator.wikimedia.org/T402958][https://phabricator.wikimedia.org/T396789]
* Upon clicking an empty search bar, logged-out users will see suggestions of articles for further reading on English Wikipedia beginning the week of September 22. The feature will be available on both desktop and mobile. All non-English wikis received this change in June and July. The goal is to make it easier for users to find articles. [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments/Search Suggestions|Learn more]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:37}} community-submitted {{PLURAL:37|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.17|MediaWiki]]

'''In depth'''
* Wikifunctions now has a new capability called &quot;lightweight enumeration types&quot;, an enumeration type is simply a fixed set of values that's in the type's definition. This capability makes it quick and easy to define such a type, and allows for the reuse of values that are already present in Wikidata. Here is [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-07-19|a newsletter]] to learn more.
* The latest [[mw:Special:MyLanguage/Readers/Newsletter updates#August 2025: Newsletter #1|Readers Newsletter]] is now available. This edition includes: the formation of two new teams — Reader Growth and Reader Experience; insights into declining pageviews and account creations; highlights from the Wikimania Nairobi panel on improving the reading experience; upcoming experiments to engage new and existing readers; and more.

'''Meetings and events'''
* Spotlight on some Wikimania 2025 Sessions:
** Identifying AI-generated text by searching for ISBNs whose checksums fail: Mathias Schindler of WMDE [https://www.youtube.com/watch?v=Dw9o8Lsl974&amp;t=15910s shared tools to help communities search for these].
** [https://wikimedia.eventyay.com/talk/wikimania2025/talk/TCHZKH/ La durabilité du mouvement Wikimedia face aux défis actuels et futurs]: This session explored how Wikimedia can stay a trusted source of knowledge in the age of generative AI, information overload, and disinformation.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:50, 1 September 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29196010 --&gt;

== Tech News: 2025-37 ==

&lt;section begin=&quot;technews-2025-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/37|Translations]] are available.

'''Weekly highlight'''
* The Editing team is working on a new check: [[mw:Special:MyLanguage/Paste check|Paste check]]. This check informs newcomers who paste text into Wikipedia that the content might not be accepted. This check is an effort to increase the likelihood that the new content people are adding to Wikipedia is aligned with the Movement's commitment to offering information under a free content license. This check will soon be tested at a few wikis. If your community is interested in this test, please [[phab:T403680|tell us in this task]], or [[mw:Talk:Edit check|contact the team]].

'''Updates for editors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Later this week, users of the &quot;{{int:codemirror-beta-feature-title}}&quot; [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] will be able to use a [[w:en:Lint (software)|linting tool]] to see errors or other potential problems in wikitext in real time. See the [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Linting|help page for more information]]. [https://phabricator.wikimedia.org/T381577]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] When browsing a wiki (like &lt;code dir=ltr&gt;en.wikipedia.org&lt;/code&gt;), the software responds in one of two ways: a desktop page, or a redirect to a mobile version on an &quot;m&quot; domain (like &lt;code dir=ltr&gt;en.m.wikipedia.org&lt;/code&gt;). Over the next three weeks, MediaWiki will start displaying the mobile version to mobile devices directly on the standard domain, without this redirect. This change does not affect existing m-dot URLs, or the &quot;Desktop view&quot; opt-out. [[mw:Requests for comment/Mobile domain sunsetting/2025 Announcement|Learn more]]. [https://phabricator.wikimedia.org/T214998]
* When an edit changes the categories of a page, the changes to the category membership counts are now happening asynchronously. This improves the speed of saving edits, especially when moving many pages to or from the same category, and reduces the risk of site outages, but it means that the counts can show outdated information for a few minutes. [https://phabricator.wikimedia.org/T365303] 
* Edits on Wikidata to qualifiers (properties and values) and references (properties and values) in a Wikidata item statement will now not add entries to the RecentChanges or Watchlist pages on all other Wikis. This is a temporary change to improve performance while other solutions are created. Wikidata's own pages remain unchanged. [[m:Wikidata For Wikimedia Projects/Reduce change propagation noise#Phase 1: Turn off (temporarily) Qualifiers and References Wikidata edits to the Recent Changes tables|Learn more]]. [https://phabricator.wikimedia.org/T401286][https://phabricator.wikimedia.org/T400698]
* Japanese-language wikis have had a major upgrade to the way that search works. The new search should generally give more accurate and more relevant search results. [https://phabricator.wikimedia.org/T318269]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.18|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:14, 9 September 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29238161 --&gt;

== Tech News: 2025-38 ==

&lt;section begin=&quot;technews-2025-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/38|Translations]] are available.

'''Updates for editors'''
* References lists that are made using the &lt;code dir=ltr&gt;&lt;nowiki&gt;&lt;references/&gt;&lt;/nowiki&gt;&lt;/code&gt; [[mw:Special:MyLanguage/Help:Cite#references-tag|tag]] will now automatically display with columns in Vector 2022 when readers are using its 'standard' settings for text-size and page-width. [https://phabricator.wikimedia.org/T334941]
* Starting in the week of October 6, on [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/small.dblist|small wikis]] and [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/medium.dblist|medium wikis]] that have the [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] enabled, all autoconfirmed users will be able to use [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] as an organizer. No changes will be made for [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/large.dblist|large wikis]] unless requested in Phabricator. This change is being made to make it easier for more people to use Event Registration, especially on wikis that are less likely to have policies related to the Event Organizer right. [[m:Special:MyLanguage/CampaignEvents/Proposal to grant autoconfirmed users on small and medium wikis the organizer access to the event registration tool|Learn more]].
* Users that search using regular expressions (regex) can now use additional features including:
** for the &lt;code dir=ltr&gt;intitle:&lt;/code&gt; keyword: [[mw:Special:MyLanguage/Help:CirrusSearch#Metacharacters|metacharacters]] for start-of-line (&lt;code dir=ltr&gt;^&lt;/code&gt;) and end-of-line (&lt;code dir=ltr&gt;$&lt;/code&gt;) anchors [https://phabricator.wikimedia.org/T317599]
** for both &lt;code dir=ltr&gt;intitle:&lt;/code&gt; and &lt;code dir=ltr&gt;insource:&lt;/code&gt; keywords: shorthand [[mw:Special:MyLanguage/Help:CirrusSearch#Character_Classes|character classes]] for digits (&lt;code dir=ltr&gt;\d&lt;/code&gt;), whitespace (&lt;code dir=ltr&gt;\s&lt;/code&gt;), and word characters (&lt;code dir=ltr&gt;\w&lt;/code&gt;); and [[mw:Special:MyLanguage/Help:CirrusSearch#Escape codes|escape codes]] for line feed (&lt;code dir=ltr&gt;\r&lt;/code&gt;), newline (&lt;code dir=ltr&gt;\n&lt;/code&gt;), tab (&lt;code dir=ltr&gt;\t&lt;/code&gt;), and unicode (e.g. &lt;code dir=ltr&gt;\uHHHH&lt;/code&gt;). [https://phabricator.wikimedia.org/T403212]
* When you search for text that looks like an IP, the system will now show search results. It used to take you to the contributions for that IP instead of showing search results. [https://phabricator.wikimedia.org/T306325]
* [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on September 24. This is planned at [https://zonestamp.toolforge.org/1758726000 15:00 UTC]. This is for the datacenter server switchover backup tests which happen twice a year. You can [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|read more about the background and details of this process on the Diff blog]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was fixed that affected users who used the page-tabs to switch from wikitext editing of a section into the visualeditor. [https://phabricator.wikimedia.org/T401043]

'''Updates for technical contributors'''
* The MediaWiki Interfaces team is redesigning the Wikimedia REST API Sandbox with Codex. If you have feedback on improvements for the API documentation or what makes developer experiences smooth (or frustrating), you’re invited to [https://calendar.google.com/calendar/u/0/appointments/schedules/AcZssZ2aZzbXeQvjOF7gB1fJXiwAYemQjKf4sXNaRODPA7_obFyNBwkzNkoVCoTF-aeov89kIjXHbCQm join an upcoming discovery interview], or [[mw:MediaWiki Interfaces Team/Developer Feedback/Wikimedia Web APIs|leave feedback onwiki]]. [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/C4FBAOA57PH6G5ORVMAUF5TGYBLZDU5Q/|Learn more]].
* Edits to Wikidata aliases (an alternative name for an item or a property) will now be shown in RecentChanges and Watchlist entries on other wikis less often, reducing unnecessary notifications. This will reduce the overall quantity of 'noisy' entries. Wikidata's own pages remain unchanged. [[m:Wikidata For Wikimedia Projects/Reduce change propagation noise#Phase 1: More granular Alias tracking|Learn more]]. [https://phabricator.wikimedia.org/T401288]
* The new [https://www.unicode.org/versions/Unicode17.0.0/ Unicode 17.0] version has been released. The [[:c:Category:Unicode Module Datasets|datasets on Commons]] for the [[:d:Q39301585|Module:Unicode data]] have been updated. Wikipedias that do not use the Commons datasets should either update their own data or switch to the Commons datasets.
* Users of the [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] Structured Contents endpoints can now access [https://enterprise.wikimedia.com/blog/parsed-wikipedia-tables/ Parsed Tables]. The new Parsed Tables feature extracts and represents Wikipedia tables in structured JSON. This improves machine accessibility as part of the [https://enterprise.wikimedia.com/api/structured-contents/ Structured Contents initiative]. Structured Contents output is freely available through the [https://enterprise.wikimedia.com/docs/on-demand/#article-structured-contents-beta On-demand API], or through Wikimedia Cloud Services. 
* A [https://www.kaggle.com/datasets/wikimedia-foundation/english-wikipedia-people-dataset dataset of English Wikipedia biographical information] from [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] has been published on Kaggle, for evaluation and research. This provides structured data from more than 1.5 million biographies, including birth and death dates, education, affiliations, careers, awards, and more (from a June 2024 snapshot).
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.19|MediaWiki]]

'''Meetings and events'''
* [[wmania:Special:MyLanguage/2026:Scholarships|Scholarship applications]] for Wikimania 2026 in Paris, France, are open until October 31.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:07, 15 September 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29263921 --&gt;

== Tech News: 2025-39 ==

&lt;section begin=&quot;technews-2025-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/39|Translations]] are available.

'''Weekly highlight'''
* [https://zonestamp.toolforge.org/1758726000 On September 24th at 15:00 UTC], all Wikimedia sites users will experience a brief read-only period due to a scheduled [[m:Special:MyLanguage/Tech/Server switch|datacenter server switchover]]. The Wikimedia Foundation's Site Reliability Engineering (SRE) team will redirect all traffic from one primary server to its backup. You can listen to the switchover using the [http://listen.hatnote.com/ &quot;Listen to Wikipedia&quot;] tool, where you will hear edits stop for a few minutes during the read-only phase, then resume. This twice-yearly datacenter server switchover ensures reliability by testing the backup datacenter, so that our sites can stay online even if the primary datacenter fails. You can [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|read more about the process on the Diff blog]].

'''Updates for editors'''
* Editors of [[f:Special:Mylanguage/Wikifunctions:Status updates/2025-09-12#Next round of Wiktionaries to receive embedded Wikifunctions calls|60 more Wiktionaries]] will soon be able to call [[f:Special:MyLanguage/Wikifunctions:Introduction|functions from Wikifunctions]] and integrate them into their pages. A function takes one or more inputs and transforms them into a desired output, like adding numbers, converting miles to meters, calculating elapsed time, or declining a word into a case. They will join the other [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-08-29#Wikifunctions available on 65 Wiktionaries|65 Wiktionary language editions]], which already have access to embedded Wikifunctions calls. Later this year, plans are in place to expand to more Wiktionaries and the Incubator.
* A new [[mw:Special:MyLanguage/Help:Magic words#Technical metadata of another page|parser function]] has been added: &lt;code&gt;&lt;nowiki&gt;{{#contentmodel}}&lt;/nowiki&gt;&lt;/code&gt;. Template editors and admins can use it to get the localized or canonical name of the [[mw:Special:MyLanguage/Help:ChangeContentModel|content model]] of a specific page. The function makes it easier to create and edit system messages, such as ''MediaWiki:editinginterface'', even when you switch types of pages, like wiki, JavaScript, CSS or JSON page. [https://phabricator.wikimedia.org/T328254]
* Adding or editing a &lt;code&gt;DISPLAYTITLE&lt;/code&gt; for an article using VisualEditor will no longer be broken. Editors who use VisualEditor mode to modify the &lt;code&gt;&lt;nowiki&gt;{{DISPLAYTITLE}}&lt;/nowiki&gt;&lt;/code&gt; would no longer have the literal text &quot;DISPLAYTITLE&quot; or its localized variant added to their articles. A list of pages that may have been affected and might need cleanup is documented in [[phab:P83438|this ticket]]. 
* Beta users of the Wikipedia Android app can now try the redesigned [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Activity Tab Experiment|Activity tab]], which replaces the Edits tab. The new tab offers personalized insights into reading, editing, and donation activity, while simplifying navigation and making app use more engaging.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:12}} community-submitted {{PLURAL:12|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.20|MediaWiki]]

'''In depth'''
* Wikifunctions users can now import many essential facts involving [[f:Special:MyLanguage/Z6011|geo-coordinates]], [[f:Special:MyLanguage/Z6010|quantities]] and [[f:Special:MyLanguage/Z6064|time]] values from Wikidata. This is made possible by the creation of Wikifunctions types for these values, which makes them available for use by functions in Wikifunctions. Learn more about how this works in [[c:File:ImportingWikidataDatatypesIntoWikifunctions.webm|this video]] and Wikifunctions' [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-08-01#News in Types I: Wikidata quantity|August 1 newsletter]] (for quantities) and [[f:Special:MyLanguage/Wikifunctions:Status updates/2025-08-22#News in Types: Wikidata geo-coordinate|August 22 newsletter]] (for geo-coordinates).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:55, 22 September 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29305556 --&gt;

== Tech News: 2025-40 ==

&lt;section begin=&quot;technews-2025-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/40|Translations]] are available.

'''Weekly highlight'''
* A major software upgrade has been made to [[phab:|Phabricator]]. The update introduces performance improvements, a refreshed search interface, enhancements to Maniphest task search, updates to user profile pages and project workboards, new Herald automation features, as well as general text input, mobile experience improvements and more. [https://phabricator.wikimedia.org/phame/post/view/321/iterative_improvements_september_2025/]

'''Updates for editors'''
* The Community Tech team will release the new Community Wishlist extension on October 1, that will improve the way wishes will be submitted. The new extension will allow users to add tags to their wishes to better categorise them, and (in a future iteration) to filter them by status, tags and focus areas. It will also be possible to support individual wishes again, as requested by the community in many instances. The old system will be retired. There will be a brief period of downtime while the extension is deployed and wishes are migrated to the new system. You can read more about this [[:m:Special:MyLanguage/Community Wishlist/Updates|in the latest update]] or you can consult the [[:mw:Special:MyLanguage/Help:Extension:CommunityRequests|current documentation on MediaWiki]].
* As announced [[diffblog:2025/09/02/better-detecting-bots-and-replacing-our-captcha/|on Diff blog]], the production trial of the [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/hCaptcha|hCaptcha]] service for bot detection has begun. The trial is currently using hCaptcha to protect account creation on Chinese, Persian, Portuguese, Indonesian, Japanese, and Turkish Wikipedias, where it will replace our existing [[mw:Special:MyLanguage/Extension:ConfirmEdit#FancyCaptcha|CAPTCHA]] (FancyCaptcha). The goal with the trial is to better block bots while also improving usability and accessibility for users who encounter CAPTCHA challenges.
* The [[mw:Special:MyLanguage/Extension:CampaignEvents|CampaignEvents]] extension has been [[m:Special:MyLanguage/CampaignEvents/Deployment status|deployed]] to Wikimedia Commons. The extension makes it easier to organize and participate in collaborative activities, like edit-a-thons and WikiProjects, on the wikis. On Commons, anyone who is a registered user can use it as an event participant. To use it as an organizer, someone needs to have the [[c:Special:MyLanguage/Commons:Event organizers|event organizer right]].
* [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new feature to re-use references with different details has been released to German Wikipedia. You can [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#test|test the feature]] on testwiki or [https://en.wikipedia.beta.wmcloud.org/wiki/Sub-referencing on betawiki] as well. Please share your thoughts on [[:m:Talk:WMDE Technical Wishes/Sub-referencing#Templates used in sub-references|using templates in sub-references]] or [[:m:Talk:WMDE Technical Wishes/Sub-referencing#Pilot wikis|volunteer to become a pilot wiki]].
* On wikis using the [[mw:Special:MyLanguage/Help:Growth/Mentorship|Mentorship]] system, communities can now opt experienced editors out of Mentorship through [[{{#special:CommunityConfiguration/Mentorship}}]]. Within this setting, communities may define thresholds, based on edit count and account age, to decide when an editor is considered experienced enough to no longer receive Mentorship. [https://phabricator.wikimedia.org/T403563]
* The Editing Team and the Machine Learning Team are working on a new check for newcomers: [[mw:Special:MyLanguage/Edit check/Tone Check|Tone check]]. Using a prediction model, this check will encourage editors to improve the tone of their edits, using artificial intelligence. We invite volunteers to review the first version of the Tone language model for the following languages: Arabic, Czech, German, Hebrew, Indonesian, Dutch, Polish, Russian, Turkish, Chinese, Farsi, Italian, Norwegian, Romanian and Latvian. Users from these wikis interested in reviewing this model are [[mw:Special:MyLanguage/Edit_check/Tone_Check/Model_evaluation|invited to sign up at MediaWiki.org]]. The deadline to sign up is on October 3, which will be the start date of the test.
* The rollout of [[:mw:Special:MyLanguage/Help:Manage blocks|multiblocks]] had the side effect that non-active block logs may have been shown on {{#special:Contributions}} and on blocked users' user and user_talk pages. This issue will be fully resolved in a few days. As part of the fix, [{{fullurl:Special:Allmessages|prefix=sp-contributions-blocked-notice}} messages prefixed with &lt;code&gt;sp-contributions-blocked-notice&lt;/code&gt;] will be removed and replaced with [{{fullurl:Special:Allmessages|prefix=blocked-notice-logextract}} those prefixed with &lt;code&gt;blocked-notice-logextract&lt;/code&gt;] in a few weeks. Please help translate the new messages and update any local overrides if needed.
* There was a bug with links added using visual editor if they included characters such as &lt;code dir=ltr&gt;&lt;nowiki&gt;[ ] |&lt;/nowiki&gt;&lt;/code&gt; after the fragment identifier (&lt;code&gt;&lt;nowiki&gt;#&lt;/nowiki&gt;&lt;/code&gt;). They were not encoded properly creating an incorrect link. This has been fixed. [https://phabricator.wikimedia.org/T404823]
* One new wiki has been created: a {{int:project-localized-name-group-wikiquote/en}} in [[d:Q9237|Malay]] ([[q:ms:|&lt;code&gt;q:ms:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T404698]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/User Info|User Info Card]] now displays currently active global lock/blocks. [https://phabricator.wikimedia.org/T401128]

'''Updates for technical contributors'''
* Later this week, editors using Lua modules will be able to use the &lt;code&gt;[[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#mw.title.newBatch|mw.title.newBatch]]&lt;/code&gt; function to look up the existence of up to 25 pages at once, in a way that only increases the [[mw:Special:MyLanguage/Manual:Parser functions#Expensive parser functions|expensive function]] count once.
* A new [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group|Unsupported Tools Working Group]] has been formed as part of ongoing efforts to collectively determine technical work priorities, similar to the [[m:Special:MyLanguage/Product and Technology Advisory Council|Product &amp; Technology Advisory Council]] (PTAC). The working group will help prioritize and review requests for support of unmaintained extensions, gadgets, bots, and tools. For the first cycle, the group will be prioritizing an unsupported Wikimedia Commons tool.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.21|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:52, 29 September 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29355230 --&gt;

== Tech News: 2025-41 ==

&lt;section begin=&quot;technews-2025-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/41|Translations]] are available.

'''Weekly highlight'''
* [[mw:Special:MyLanguage/Help:Edit check#paste|Paste Check]] is a new Edit Check feature to help avoid and fight copyright violations. When editors paste text into an article, Paste Check prompts them to confirm the origin and licensing of the content. Starting Wednesday, 8 October, [[phab:T403680|22 wikis will test Paste Check]]. Paste Check will help new volunteers understand and follow the policies and guidelines necessary to make constructive contributions to Wikipedia projects.
'''Updates for editors'''
* Mobile devices will receive mobile articles directly on the standard domain (like &lt;code&gt;en.wikipedia.org&lt;/code&gt;), instead of via a redirect to an &quot;m&quot; domain (like &lt;code&gt;en.m.wikipedia.org&lt;/code&gt;). This change improves performance. This week it will be enabled on Wikipedias. The existing mobile URLs and the &quot;Desktop view&quot; opt-out remain available. [[mw:Requests for comment/Mobile domain sunsetting/2025 Announcement|Learn more]]. [https://phabricator.wikimedia.org/T214998]
* New [[mw:Special:MyLanguage/Help:CirrusSearch#creationdate and lasteditdate|date filters]], &lt;code dir=ltr&gt;creationdate:&lt;/code&gt; and &lt;code dir=ltr&gt;lasteditdate:&lt;/code&gt;, are now available in the wiki search engine. This allows users to filter search results by a page's first or last revision date. The filters support comparison operators (e.g. &lt;code dir=ltr&gt;&gt;2024&lt;/code&gt;) and relative dates (e.g. &lt;code dir=ltr&gt;today-1d&lt;/code&gt;), making it easier to find recently updated content or pages within specific age ranges. [https://phabricator.wikimedia.org/T403593]
* [[f:|Wikifunctions]] now supports rich text in embedded calls across the 150 wikis where it's enabled. To showcase this, the team created a [[f:Z26333|Latin declination table]] that Wiktionary editors can use to automatically generate noun forms, producing clear, formatted results — see an [[f:Wikifunctions:Embedded function calls/Wiktionary tables demonstration|example output]]. If you need any help or have any feedback, please [[f:Wikifunctions:Project chat|contact the Wikifunctions Team]]. [https://phabricator.wikimedia.org/T397402]
* An edit link will now appear inside the categories box on article pages for logged in users, which will directly launch the VisualEditor category dialog. [https://phabricator.wikimedia.org/T291691]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:34}} community-submitted {{PLURAL:34|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a problem downloading pdf files last week and that has been resolved. [https://phabricator.wikimedia.org/T405957]
'''Updates for technical contributors'''
* The field &lt;code dir=ltr&gt;rev_sha1&lt;/code&gt; in the revision database table is being removed in favor of &lt;code dir=ltr&gt;content_sha1&lt;/code&gt; in the content database table. See [https://lists.wikimedia.org/hyperkitty/list/cloud@lists.wikimedia.org/thread/2D2M3SP4WHR6BXXKTZ2PBLZQYR3EGQVR/ the announcement] for more information.
* The [[mw:Special:MyLanguage/Reading/Web|Reader Experience team]] will roll out [[w:en:Light-on-dark color scheme|Dark Mode]] user interface on all Wikimedia sites on October 29, 2025. All anonymous users of Wikimedia sites will have the option to activate a color scheme that features light-colored text on a dark background. This is designed to provide a more comfortable reading experience, especially in low-light situations. Template authors and technical contributors are encouraged to [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-04|learn how to make pages ready for Dark mode]] and address any compatibility issues found in templates in their wiki before the enablement. Please contact the Web team for questions or any support on [[mw:Talk:Reading/Web/Accessibility for reading#|this talk page]] before the enablement. [https://phabricator.wikimedia.org/T395628]
* Starting on Monday, October 6, API endpoints under the &lt;code&gt;rest.php&lt;/code&gt; path will be rerouted through a new internal API Gateway. Individual wikis will be updated based on the standard release groups, with total traffic increased over time. This change is expected to be non-breaking and non-disruptive. If any issues are observed, please file a Phabricator ticket to the [[phab:tag/serviceops/|Service Ops team board]]. [https://phabricator.wikimedia.org/T400130]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.22|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W41&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:23, 6 October 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29400897 --&gt;

== Tech News: 2025-42 ==

&lt;section begin=&quot;technews-2025-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/42|Translations]] are available.

'''Weekly highlight'''
* Last week, improvements to account security and two-factor authentication (2FA) features were enabled across all wikis. These changes include user interface improvements for [https://auth.wikimedia.org/metawiki/wiki/Special:AccountSecurity Special:AccountSecurity], the support of multiple 2FA methods via authenticator apps and portable security keys (previously users could only enable one method), and a new Recovery Codes module which facilitates fewer account lockouts due to lost two-factor apps and devices. As part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] project, work is continuing through the rest of 2025 on further user experience improvements, and support for passkeys as an alternate second factor. 

'''Updates for editors'''

* Another part of the Account security project is making 2FA generally available to all users. Along with editors with advanced privileges, such as administrators and bureaucrats, 40% of editors now have access to 2FA. You can check if you have access at [https://auth.wikimedia.org/metawiki/wiki/Special:AccountSecurity Special:AccountSecurity]. Instructions for activation are on the linked page. The plan is to continue increasing availability if it is determined that the user support capabilities are able to support global usage. [https://phabricator.wikimedia.org/T400579]
* This week, users at wikis where talk page [[mw:Special:MyLanguage/Talk pages project/Usability|Usability Improvements]] are already available by default (everywhere ''except'' the 12 wikis listed in [[phab:T379264|T379264]]) will gain the ability to Thank a comment directly from the talk page it appears on. Before this change, Thanking could only be done by visiting the revision history of the talk page. You can [[diffblog:2025/10/13/revolutionizing-gratitude-a-new-era-of-thanking-comments/|learn more about this change]]. [https://phabricator.wikimedia.org/T366095]
* Users who have not [[Special:Preferences#mw-prefsection-personal-email|verified their email address]] will soon be receiving monthly Notification reminders to do so. This is because users who have verified their email can more easily recover their account. These reminders will not be sent if the user is inactive or removes the unverified email from their account. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Email_confirmation][https://phabricator.wikimedia.org/T58074]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a fix was made for an occasional error with saving translated paragraphs in the Content Translation tool, and the related error messages are now easier to see. [https://phabricator.wikimedia.org/T376531]

'''Updates for technical contributors'''

* The Unsupported Tools Working Group has chosen [[c:Special:MyLanguage/Commons:Video2commons|Video2Commons]] as the first tool for its pilot cycle. The group will explore ways to improve and sustain the tool over the coming months. [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group|Learn more on Meta]].
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.23|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:59, 13 October 2025 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29434481 --&gt;

== Tech News: 2025-43 ==

&lt;section begin=&quot;technews-2025-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/43|Translations]] are available.

'''Updates for editors'''
* To optimize how user data is stored in our databases, the saved preferences of users who haven't logged in for over five years and have fewer than 100 edits will be cleared. When those users return, default settings will apply. [https://phabricator.wikimedia.org/T406724]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, there was a broken link from the GlobalContributions interface message to the XTools GlobalContributions page which has now been fixed. [https://phabricator.wikimedia.org/T406415]

'''Updates for technical contributors'''
* The work to reroute all traffic to API endpoints under the &lt;code dir=ltr&gt;&lt;nowiki&gt;rest.php&lt;/nowiki&gt;&lt;/code&gt; route through a common API gateway is now complete. If any issues are observed, please file a phabricator ticket to the [[phab:tag/serviceops/|Service Ops team board]].
* Edits to Wikidata references or qualifiers will now be shown in RecentChanges and Watchlist entries on other wikis less often, reducing unnecessary notifications. This will reduce the overall quantity of 'noisy' entries. Wikidata's own pages remain unchanged. [https://phabricator.wikimedia.org/T401290]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.24|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:36, 20 October 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29478670 --&gt;

== Tech News: 2025-44 ==

&lt;section begin=&quot;technews-2025-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/44|Translations]] are available.

'''Updates for editors'''
* The Wikipedia iOS app has launched an A/B/C test of improvements made to the tabbed browsing feature for select regions and languages. The test, named “More dynamic tabs”, explores new tab experiences and includes “Did you know” and “Because you read” article recommendations. You can [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Tabbed Browsing (Tabs)/New Tab Experience and Recommendations Experiment|read more on the project page]].
* Autoconfirmed users on [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/small.dblist|small]] and [[gitiles:operations/mediawiki-config/+/a2d2aaab9ace84280dd2f4c70a33bb69cd73850f/dblists/medium.dblist|medium wikis]] with the CampaignEvents extension can now use [[m:Special:MyLanguage/Event Center/Registration|Event Registration]] without the Event Organizer right. This feature lets organizers enable registration, manage participants, and lets users register with one click instead of signing event pages.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue of flashing colors when holding or pressing the arrow keys under the dark mode settings in Vector 2022 has been fixed. [https://phabricator.wikimedia.org/T402285]

'''Updates for technical contributors'''

* The CampaignEvents extension will be deployed to all remaining wikis during the week of 17 November 2025. The extension currently includes three features: Event Registration, Collaboration List, and Invitation List. For this rollout, Invitation List will not be enabled on Wikifunctions and MediaWiki unless requested by those communities. [[m:Special:MyLanguage/CampaignEvents/Deployment status|Visit the deployment page to learn more]].
* The SwaggerUI-based REST sandbox experience is now live on all wiki projects. The sandbox can be accessed through the [[{{#special:RestSandbox}}]] page. Please report any issues to the MediaWiki Interfaces team board, or join the discussion on the [[mw:Special:MyLanguage/MediaWiki Interfaces Team/Feature Feedback/REST Sandbox|project launch]] page. [https://phabricator.wikimedia.org/project/board/6931/]
* Transform endpoints with a trailing slash path in the MediaWiki REST API are now marked as deprecated. They will remain functional during this time, but removal is expected by the end of January 2026. All API users currently calling them are encouraged to transition to the non-trailing slash versions. Both endpoint variations can be found and tested using the [https://test.wikipedia.org/w/index.php?api=mw-extra&amp;title=Special%3ARestSandbox REST Sandbox]. See the [[mw:API/Deprecation|MediaWiki REST API Deprecation]] page for more detailed information about the API deprecation policies and procedures.
* A dedicated [[mw:API:REST API/Changelog|changelog now exists for the MediaWiki REST API]]. The changelog provides an overview of these changes, making it easier for developers to keep track of improvements and iterations. Announcements will also continue to flow through the standard communication channels, including Tech News and email distribution lists, but can now be more easily referenced from a central location. If you have feedback about the style, structure, or content of this changelog, please [[mw:API talk:REST API/Changelog|join the discussion]].
* Administrators can delete the tracking category which was previously added by the JsonConfig extension, as it is no longer used. See the categories linked from [[d:Q130635582#sitelinks-wikipedia|Q130635582]]. It is OK if there are still pages listed in the category as that is just a caching issue, and they will be automatically cleared out the next time each page is edited. [https://phabricator.wikimedia.org/T378352]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.25|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:31, 27 October 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29513638 --&gt;

== Tech News: 2025-45 ==

&lt;section begin=&quot;technews-2025-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/45|Translations]] are available.

'''Updates for editors'''
* Administrators will now find that [[{{#special:MergeHistory}}]] is now significantly more flexible about what it can merge. It can now merge sections taken from the middle of the history of the source (rather than only the start) and insert revisions anywhere in the history of the destination page (rather than only the start). [https://phabricator.wikimedia.org/T382958]
* For users with &quot;{{int:discussiontools-preference-autotopicsub}}&quot; [[Special:Preferences#mw-prefsection-editing|enabled in their preferences]], starting a new topic or adding a reply to an existing topic will now subscribe them to replies to that topic. Previously, this would only happen if the DiscussionTools &quot;{{int:Skin-action-addsection}}&quot; or &quot;{{int:Discussiontools-replybutton}}&quot; widgets were used. When DiscussionTools was originally launched existing accounts were not opted in to automatic topic subscriptions, so this change should primarily affect newer accounts and users who have deliberately changed their preferences since that time. [https://phabricator.wikimedia.org/T290778]
* Scribunto modules can now be used to [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#SVG library|generate SVG images]]. This can be used to build charts, graphics and other visualizations dynamically through Lua, reducing the need to compose them externally and upload them as files. [https://phabricator.wikimedia.org/T405861]
* Wikimedia sites now provide all anonymous users with the option to enable a dark mode color scheme, featuring light-colored text on a dark background. This enhancement aims to deliver a more enjoyable reading experience, especially in dimly lit environments.  [https://phabricator.wikimedia.org/T395628]
* Users with large watchlists have long faced timeouts when editing [[Special:EditWatchlist|Special:EditWatchlist]]. The page now loads entries in smaller sections instead of all at once due to a paging update, allowing everyone to edit their watchlists smoothly. As part of the database update, sorting by expiry has been removed because it was over 100× slower than sorting by title. A [https://meta.wikimedia.org/wiki/Community_Wishlist/W454 community wish] has been created to explore alternative ways to restore sort-by-expiry. If this feature is important to you, please support the wish! [https://phabricator.wikimedia.org/T41510]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:31}} community-submitted {{PLURAL:31|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the fixing of the persisting highlighting when using VisualEditor find and replace during a query. [https://phabricator.wikimedia.org/T407318]
'''Updates for technical contributors'''
* Since 2019 the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]] at https://w.wiki is available for all Wikimedia wikis to create short links to articles, permalinks, diffs, etc. It is available in the sidebar as &quot;Get shortened URL&quot;. There are 30 wikis that also install an older &quot;ShortUrl&quot; extension. The old extension will soon be removed. This means &lt;code&gt;/s/&lt;/code&gt; URLs will not be advertised under article titles via HTML &lt;code dir=ltr&gt;class=&quot;title-shortlink&quot;&lt;/code&gt;. The &lt;code&gt;/s/&lt;/code&gt; URLs will keep working. [https://phabricator.wikimedia.org/T107188]
* On Thursday, October 30, the [[:mw:Special:MyLanguage/MediaWiki Interfaces Team|MediaWiki Interfaces]] and [[:mw:Special:MyLanguage/Wikimedia Site Reliability Engineering|SRE Service Operations]] teams began rerouting Action API traffic through a common API gateway. Individual wikis will be updated based on the standard release groups, with total traffic increased over time. This change is expected to be non-breaking and non-disruptive. If any issues are observed, please file a Phabricator ticket to the [https://phabricator.wikimedia.org/tag/serviceops/ Service Ops team] board.
* MediaWiki Train deployments will pause for the final two weeks of 2025: 22 December and 29 December. Backport windows will also pause between Monday, 22 December 2025 and Thursday, 2 January 2026. A backport window is a scheduled time to add things like bug fixes and configuration changes. There are seven deployment trains remaining for 2025. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/SMWTEAES4SDLDUSK4HMWNBSKNCXZAWYN/]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.45/wmf.26|MediaWiki]]

'''In depth'''
* In 2025, the Wikimedia Foundation reported that AI systems and search engines increasingly use Wikipedia content without driving users to the site, contributing to an 8% drop in human pageviews compared to 2024. After detecting bots disguised as humans, Wikimedia updated its traffic data to reflect this shift. Read more about current user trends on Wikipedia in [[diffblog:2025/10/17/new-user-trends-on-wikipedia/|a Diff blog post]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:34, 3 November 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29552512 --&gt;

== Tech News: 2025-46 ==

&lt;section begin=&quot;technews-2025-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/46|Translations]] are available.

'''Updates for editors'''
[[File:Talk pages default look (April 2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* Starting November 12, users will see a change in the [[m:Special:MyLanguage/Talk pages project/Feature summary#Usability improvements|appearance of talk pages]] on [[Phab:T379264|some Wikipedias]]. Almost [[phab:T392121|all wikis]] have received this design change; [[phab:T409297|English Wikipedia]] will get these changes later. You can read more [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. Users can opt out of these changes [[Special:Preferences#mw-prefsection-editing|in their user preferences]] in &quot;{{int:discussiontools-preference-visualenhancements}}&quot;. [https://phabricator.wikimedia.org/T379264]
* MediaWiki can now display a [[mw:Special:MyLanguage/Help:Protection indicators|page indicator]] automatically while a page is protected. This feature is disabled by default. It can be enabled by [[m:Special:MyLanguage/Requesting wiki configuration changes|community request]]. [https://phabricator.wikimedia.org/T12347]
* Using the &quot;{{int:showpreview}}&quot; or &quot;{{int:showdiff}}&quot; buttons in the wikitext editor will now carry over certain URL parameters like '[[mw:Special:MyLanguage/Manual:Parameters to index.php#useskin|useskin]]', '[[mw:Special:MyLanguage/Manual:Parameters to index.php#uselang|uselang]]' and '[[mw:Special:MyLanguage/Help:Section#Editing sections|section]]'. This update also fixes an issue where, if the browser crashed while previewing an edit to a single section, saving this edit could overwrite the entire page with just that section’s content. [https://phabricator.wikimedia.org/T62744][https://phabricator.wikimedia.org/T24029][https://phabricator.wikimedia.org/T155097]
* Wikivoyage wikis can use [[mw:Special:MyLanguage/Help:Extension:Kartographer#Markers and counters|colored map markers in the article text]]. The text of these markers will now be shown in contrasting black or white color, instead of always being white. Local workarounds for the problem can be removed. [https://phabricator.wikimedia.org/T369454]
* The Activity tab in the Wikipedia Android app is now available for all users. The new tab offers personalized insights into reading, editing, and donation activity, while simplifying navigation and making app use more engaging. [https://www.mediawiki.org/wiki/Wikimedia_Apps/Team/Android/Activity_Tab_Experiment]
* The Reader Growth team is launching an experiment called &quot;Image browsing&quot; to test how to make it easier for readers to browse and discover images on Wikipedia articles. This experiment, a mobile-only A/B test, will go live on English Wikipedia in the week of November 17 and will run for four weeks, affecting 0.05% of users on English wiki. The test launched on November 3 on Arabic, Chinese, French, Indonesian, and Vietnamese wikis, affecting up to 10% of users on those wikis. [https://www.mediawiki.org/wiki/Readers/Reader_Growth/WE3.1.3_Image_Browsing]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example the inability to lock accounts on mobile sites has been fixed. [https://phabricator.wikimedia.org/T256185]

'''Updates for technical contributors'''
* [[wikitech:Help talk:Toolforge/Toolforge standards committee#November 2025 committee nominations|Nominations are open on Wikitech]] for new [[wikitech:Help:Toolforge/Toolforge standards committee|Toolforge standards committee]] members. The committee oversees the Toolforge [[wikitech:Help:Toolforge/Right to fork policy|Right to fork policy]] and [[wikitech:Help:Toolforge/Abandoned tool policy|Abandoned tool policy]] among other duties. Nominations will remain open through 2025-11-28.
* The [[w:JSON Web Token#Standard fields|JWT issuer field]] in [[mw:Special:MyLanguage/OAuth/For Developers#OAuth 2|OAuth 2 access tokens]] for [[m:Special:MyLanguage/Help:Unified login|SUL wikis]] has been changed to &lt;code&gt;&lt;nowiki&gt;https://meta.wikimedia.org&lt;/nowiki&gt;&lt;/code&gt;. Old access tokens will still work. [https://phabricator.wikimedia.org/T399199]
* The [[w:JSON Web Token#Standard fields|JWT subject field]] in [[mw:Special:MyLanguage/OAuth/For Developers#OAuth 2|OAuth 2 access tokens]] will soon change from &lt;code&gt;&lt;user id&gt;&lt;/code&gt; to &lt;code dir=ltr style=&quot;white-space:nowrap&quot;&gt;mw:&lt;identity type&gt;:&lt;user id&gt;&lt;/code&gt;, where &lt;code&gt;&lt;identity type&gt;&lt;/code&gt; is typically &lt;code dir=ltr&gt;CentralAuth:&lt;/code&gt;&lt;!-- not a typo --&gt; (for [[m:Special:MyLanguage/Help:Unified login|SUL wikis]]) or &lt;code dir=ltr style=&quot;white-space:nowrap&quot;&gt;local:&lt;wiki id&gt;&lt;/code&gt; (for other wikis). This is to avoid conflicts between different user ID types, and to make OAuth 2 access tokens and the &lt;code&gt;sessionJwt&lt;/code&gt; cookie more similar. Old access tokens will still work. [https://phabricator.wikimedia.org/T399199]
* MediaWiki's block messages ([[MediaWiki:Blockedtext|blockedtext]], [[MediaWiki:Blockedtext-partial|blockedtext-partial]], [[MediaWiki:Autoblockedtext|autoblockedtext]], [[MediaWiki:Systemblockedtext|systemblockedtext]], [[MediaWiki:Blockedtext-tempuser|blockedtext-tempuser]], [[MediaWiki:Autoblockedtext-tempuser|autoblockedtext-tempuser]]) now support additional parameters indicating whether the user is blocked from editing their own user talk page &lt;code&gt;&lt;nowiki&gt;$9&lt;/nowiki&gt;&lt;/code&gt; or emailing other users &lt;code&gt;&lt;nowiki&gt;$&lt;/nowiki&gt;&lt;nowiki&gt;10&lt;/nowiki&gt;&lt;/code&gt;. [https://phabricator.wikimedia.org/T285612]
* A &lt;code&gt;REL1_45&lt;/code&gt; branch for MediaWiki core and each of the extensions and skins in Wikimedia git has been created. This is the first step in the release process for MediaWiki 1.45.0, scheduled for late November 2025. If you are working on a critical bug fix or working on a new feature, you may need to take note of this change. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/ZUY7TY3Z6XPZWZVAZV63OPO5OW52Q6GE/]
* The process for generating CirrusSearch dumps has been updated due to slowing performance. If you encounter any issues migrating to the replacement dumps, please contact the Search Platform Team for support. [https://phabricator.wikimedia.org/T366248][https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/3KQPOR6ACVN6OVLMLZPIBXQSWQKW4E3K/]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.2|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:38, 10 November 2025 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29606150 --&gt;

== Tech News: 2025-47 ==

&lt;section begin=&quot;technews-2025-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/47|Translations]] are available.

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] is experimenting with [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4_Reading lists|reading lists on mobile web]], allowing logged-in readers with no edits to save private lists of articles for later. The experiment is running on Arabic, Chinese, French, Indonesian, and Vietnamese Wikipedias since the week of 10 November, and will begin on English Wikipedia the week of 17 November.
* Users who can’t receive their email verification code during login can now get help by submitting a form on a new special page. This update is part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] initiative. If your account has an email address, please make sure you still have access to it. When logging in from a new device or location without 2FA, you may be asked to enter a 6-digit code sent by email to finish logging in. [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security#Why are you requiring me to enter a code from my email to log in? Can I opt out of this?|Learn more]].
* One new wiki has been created: a {{int:project-localized-name-group-wikisource}} in [[d:Q13324|Minangkabau]] ([[s:min:|&lt;code&gt;s:min:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T408317]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* As part of the [[mw:Special:MyLanguage/Parsoid/Parser Unification|Parser Unification]] project, the Content Transform Team rolled out Parsoid as the default parser to many low-traffic Wikipedias and is preparing the next step to high traffic ones. This message is an invitation for you to opt-in to Parsoid, as described in the [[mw:Special:MyLanguage/Help:Extension:ParserMigration|Extension:ParserMigration]] documentation, and identify any issues you might encounter with your own workflow using bots, gadgets, or user scripts. Please, let us know through the ''&quot;Report Visual Bug&quot;'' link in the Tools sidebar or create a phab ticket and tag the [[phab:project/view/5846|Content Transform Team in Phabricator]].
* Unsupported Tools: Several issues with [[:c:Special:MyLanguage/Commons:Video2commons|Video2Commons]] have been fixed, including filename-related upload failures, black-video imports, and retry handling. AV1 support has also been added. Ongoing work focuses on backend stability, ffmpeg errors, subtitle imports, metadata handling, and playlist uploads. To track specific tasks, check the [[phab:tag/video2commons/|Phabricator board]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.3|MediaWiki]]

'''Meetings and events'''
* Save the date for the next Wikimedia Hackathon happening in Milan, Italy from May 1–3, 2026. Registration will open in January 2026. [https://pretix.eu/wikimedia/Hackathon-2026/ Scholarship applications are currently open], and will close on November 28, 2025. If you have any questions, please email &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;hackathon@wikimedia.org&lt;/bdi&gt;.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:26, 17 November 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29627455 --&gt;

== Tech News: 2025-48 ==

&lt;section begin=&quot;technews-2025-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/48|Translations]] are available.

'''Updates for editors'''
* Last week, the [[mw:Special:MyLanguage/Wikimedia Search Platform|Wikimedia Search Team]] recreated the &quot;DWIM&quot; (Do What I Mean) gadget functionality server-side, for Russian and Hebrew Wikipedias. This feature adds cross-keyboard suggestions to the standard search-box suggestions. For example, searching for ''&lt;span lang=&quot;und&quot; dir=&quot;ltr&quot;&gt;cxfcnmt&lt;/span&gt;'' on Russian Wikipedia will now add suggestions for ''&lt;span lang=&quot;ru&quot; dir=&quot;ltr&quot;&gt;счастье&lt;/span&gt;'' (&quot;happiness&quot;) that the user probably intended. They plan to enable this feature for other Russian and Hebrew wikis this week. [https://phabricator.wikimedia.org/T408734]
* Later this week, users of the &quot;{{int:codemirror-beta-feature-title}}&quot; [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] will have syntax highlighting available in [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]]. This requires that the &quot;{{int:discussiontools-preference-sourcemodetoolbar}}&quot; preference be set. [https://phabricator.wikimedia.org/T407918]
* [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|Campaign events extension]] – the set of tools for coordinating events and other on-wiki collaborations has now been deployed to all Wikimedia wikis. A new feature known as [[m:Special:MyLanguage/CampaignEvents/Collaborative contributions|Collaborative contribution]] to help organizers and participants see the impact of activities has also been added. Join the upcoming [[m:Special:MyLanguage/Event:Connection learning session 3|learning session]] to see the new feature in action and share your feedback.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug which stopped CodeReviewBot from working, has now been fixed. [https://phabricator.wikimedia.org/T410417]

'''Updates for technical contributors'''
*  Users of Wikimedia API can join a usability study to help validate the new design of Wikimedia REST API sandboxes. Interested participants should fill the [https://wikimediafoundation.limesurvey.net/487662 recruitment survey]. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IREJRRWTZTGCYWQHDMSNJFTQAEPOOAE3/]
* The MediaWiki Interfaces team is deprecating XSLT stylesheets within the Action API. Support for &lt;code dir=ltr&gt;format=xml'''&amp;xlst={stylesheet}'''&lt;/code&gt; will be removed from Wikimedia projects by the end of November, 2025. In addition, it will soon be disabled by default in MediaWiki release versions: v1.43 (LTS), v1.44, and v1.45. Support for XSLT stylesheets will be fully removed from MediaWiki v1.46 (expected to release between April and May 2026). [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/5AX7UWAVVUNUSBOIRHMNOKWOZ5EZI3JX/]
* The WDQS legacy endpoint ([https://query-legacy-full.wikidata.org/ query-legacy-full.wikidata.org]) will be decommissioned at the end of December 2025, and finally closed down on 7th January 2026. After this date, users should expect requests to query.wikidata.org that require the full graph to fail or return invalid results if they are not rewritten to use SPARQL federation. The team encourages users to ensure that tools and workflows use the supported WDQS endpoints (&lt;span dir=ltr&gt;&lt;nowiki&gt;https://query.wikidata.org/&lt;/nowiki&gt;&lt;/span&gt; - Main graph or &lt;span dir=ltr&gt;&lt;nowiki&gt;https://query-scholarly.wikidata.org/&lt;/nowiki&gt;&lt;/span&gt; - Scholarly graph). For support with migrating use cases, please review the [[d:Special:MyLanguage/Wikidata:Data_access|Data Access]] and [[d:Wikidata:Request_a_query|Request a Query]] pages for details and assistance on alternative access methods.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.4|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:56, 24 November 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29702226 --&gt;

== Tech News: 2025-49 ==

&lt;section begin=&quot;technews-2025-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/49|Translations]] are available.

'''Updates for editors'''
* The Wikipedia Year in Review 2025 will be available on December 2 for users of iOS and Android Wikipedia apps, featuring new personalized insights, updated reading highlights, and refreshed designs. Learn more on the review's [[mw:Special:MyLanguage/Wikimedia Apps/Team/Wikipedia Year in Review/Updates|project page]].
* The Growth team is working on improving the text and presentation of the Verification Email sent to new users to make them more welcoming, useful and informative. Some new text have been drafted for A/B testing and you can help by translating them. See [[phab:T396155|Phabricator]].
* [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]] will now be deployed at Japanese, Urdu and Chinese Wikipedias on December 2. Add a link is based on a prediction model that suggests links to be added to articles. While this feature has already been available on most Wikipedias, the prediction model could not support certain languages. A new model has now been developed to handle these languages, and it will be gradually rolled out to other Wikipedias over time. If you would like to know more, please contact [[mw:user:Trizek (WMF)|Trizek (WMF)]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:34}} community-submitted {{PLURAL:34|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where search boxes on some Commons pages showed no results due to switch from SpecialSearch to MediaSearch, has now been fixed. [https://phabricator.wikimedia.org/T399476]
* Two new wikis have been created:
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q36846|Toki Pona]] ([[w:tok:|&lt;code&gt;w:tok:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T404457]
** a {{int:project-localized-name-group-wikiquote}} in [[d:Q33655|Nigerian Pidgin]] ([[q:pcm:|&lt;code&gt;q:pcm:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T408318]
'''Updates for technical contributors'''
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.5|MediaWiki]]

'''In depth'''
* The Wikimedia Foundation is in the early stages of exploring approaches to '''Article guidance'''. The initiative aims to identify interventions that could help new editors easily understand and apply existing Wikipedia practices and policies when creating an article. The project is in the exploration and early experimental design phase. All community members are encouraged to [[mw:Special:MyLanguage/Article guidance|learn more]] about the project, and share their thoughts on [[mw:Special:MyLanguage/Talk:Article guidance|the talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:57, 1 December 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29732328 --&gt;

== Tech News: 2025-50 ==

&lt;section begin=&quot;technews-2025-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/50|Translations]] are available.

'''Weekly highlight'''
* Anybody who wishes to secure their user account can now use [[m:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]] (2FA). This is available to all registered users of all Wikimedia projects. This is part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] initiative. Later, 2FA will be required for all users who can take security- or privacy-sensitive actions.

'''Updates for editors'''
* Following last week's deployments, the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]] feature, which allows editors to add suggested links during editing, will be available to an additional [[Phab:T410469|33 Wikipedias]] starting on 9 December. This expansion is possible thanks to the new prediction model that now supports all languages, including those that were previously not covered. While the feature has been available on most Wikipedias for some time, this rollout brings us closer to using the improved model everywhere. If you have any questions or would like more details please contact [[mw:user:Trizek (WMF)|Trizek (WMF)]].
* Last week, the [[mw:Special:MyLanguage/Wikimedia Search Platform|Search Platform team]] added [[w:en:Transliteration|transliterated]] as-you-type search suggestions to Georgian wikis. If there are only a few regular search suggestions, then queries in Latin or Cyrillic script [[phab:T127003|are now rewritten into Georgian script]] to look for more matches. For example, searching for either &lt;bdi lang=&quot;ka-Latn&quot; dir=&quot;ltr&quot;&gt;''bedniereba''&lt;/bdi&gt; or &lt;bdi lang=&quot;ka-Cyrl&quot; dir=&quot;ltr&quot;&gt;''бедниереба''&lt;/bdi&gt; will now suggest the existing article about &lt;bdi lang=&quot;ka&quot; dir=&quot;ltr&quot;&gt;ბედნიერება&lt;/bdi&gt; (&quot;happiness&quot;). You can recommend other languages where transliterated suggestions would be useful [[phab:T375215|on Phabricator]] for future development.
* Later this week, a controlled experiment will begin for editors on the 100 largest Wikipedias who are editing a section in the mobile web visual editor. 50% of these editors will notice a new &quot;Edit full page&quot; button that will enable them to expand their editing session to the whole page. This feature is intended to make it easier for people on mobile web to edit any article section, regardless of which section-edit icon they tapped to begin. The experiment will last ~4 weeks. You can find [[phab:T409112|more details]] about the project.
* Later this week, the [[mw:Special:MyLanguage/Readers/Reader Growth|Reader Growth team]] will launch a [[mw:Special:MyLanguage/Readers/Reader Growth/WE3.1.14 Expanded Mobile Sections|mobile web experiment]] to expand all article sections by default (currently they are collapsed by default) and pin the section header the user is currently reading to the top of the page. The experiment will affect 10% of users on Arabic, Chinese, French, Indonesian, and Vietnamese Wikipedias. [https://phabricator.wikimedia.org/T409485]
* The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Wikipedia Year in Review/2025 Year in Review|Wikipedia Year in Review 2025]], a feature in the Wikipedia mobile apps (iOS and Android) that provides users with a personalised summary of their engagement with Wikipedia over the year, is now available on the iOS and Android apps. This edition includes expanded personalised insights, improved reading highlights, new donor messaging, and updated designs. Open the app to view your Year in Review and explore your reading journey from 2025.
* A recent software bug caused edits made with VisualEditor to make unintended changes to wikitext, including removing whitespace and replacing spaces with underscores in wikilinks inside citations. This was partially fixed last week, and further fixes are in progress. Editors who used VisualEditor between November 28 and December 2 should review their edits for unexpected modifications. [https://phabricator.wikimedia.org/T411238]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the incorrect handling of URLs copied from the address bar of Microsoft Edge users, has been resolved. [https://phabricator.wikimedia.org/T341281]

'''Updates for technical contributors'''
* Starting this week, users of the &quot;{{int:codemirror-beta-feature-title}}&quot; [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] will have [[mw:Special:MyLanguage/Help:Extension:CodeMirror|CodeMirror]] as the editor for Lua, JavaScript, CSS, JSON and Vue content models, instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]]. With this, the [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Linting|linters]] will be upgraded. This is part of a larger effort to eventually replace CodeEditor and provide a consistent code editing experience. [https://phabricator.wikimedia.org/T373711]
* Developers are encouraged to take the [https://wikimediafoundation.limesurvey.net/552643 2025 Developer Satisfaction Survey], which remains open until 5 January 2026. If you build software for the Wikimedia ecosystem and would like to share your experiences or feedback, your participation is greatly appreciated. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/W4WBKO6Q55UWWCCSFWQATKEXBEHP3QNR/]
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:45, 8 December 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29738112 --&gt;

== Tech News: 2025-51 ==

&lt;section begin=&quot;technews-2025-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/51|Translations]] are available.

'''Updates for editors'''
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:18}} community-submitted {{PLURAL:18|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, one of the fixes addressed an issue for temporary accounts adding an external URL, which triggered an hCaptcha request in more cases than intended, and did not display the required popup on the first attempt to publish the edit. [https://phabricator.wikimedia.org/T411927]

'''Updates for technical contributors'''
* To improve database and site performance, external links to Wikimedia projects will no longer be stored in the database. This means they will not be searchable in [[{{#special:LinkSearch}}]], will not be checked by the Spam Blacklist or AbuseFilter as new links, and will not be in the &lt;code dir=ltr&gt;externallinks&lt;/code&gt; table on database replicas. In the future this may be extended to other highly-linked trusted websites on a per-wiki basis, such as Creative Commons links on Wikimedia Commons. [https://phabricator.wikimedia.org/T405005]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.7|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:03, 15 December 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29796010 --&gt;

== Tech News: 2025-52 ==

&lt;section begin=&quot;technews-2025-W52&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2025/52|Translations]] are available.

'''Updates for editors'''
* From January, edit filters [[mw:Special:MyLanguage/Extension:AbuseFilter/Access flags|can be set]] to automatically suppress their details such as rules and list of attempted edits and actions. This will help oversighters use edit filters to prevent doxxing or other suppressible material. [https://phabricator.wikimedia.org/T290324]
* The next issue of Tech News will be sent out on 12 January 2026 because of the end of year holidays. Thank you to all of the translators, and people who submitted content or feedback, this year.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:16}} community-submitted {{PLURAL:16|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the crash that occurred when tapping &quot;First Steps&quot; in the Wikipedia Android Year in Review has now been fixed, and the feature opens as expected. [https://phabricator.wikimedia.org/T411546]

'''Updates for technical contributors'''
* Interface elements such as diffs and categories generated by MediaWiki used to have the attribute &lt;code dir=ltr&gt;data-mw=&quot;interface&quot;&lt;/code&gt; to distinguish from wiki content. The attribute has been replaced with &lt;code dir=ltr&gt;data-mw-interface=&quot;&quot;&lt;/code&gt;, to avoid potential conflicts with other &lt;code dir=ltr&gt;data-mw&lt;/code&gt; attributes, which are generated by Parsoid. [https://phabricator.wikimedia.org/T409187]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] There is no new MediaWiki version this week or next week.
'''Meetings and events'''
* The [[mw:Wikimedia Hackathon Northwestern Europe 2026|Wikimedia Hackathon Northwestern Europe 2026]] will take place on 13-14 March 2026 in Arnhem, the Netherlands. Applications just opened mid-December and will close in mid-January or earlier if capacity is reached. With space for approximately 100 participants, early application is encouraged.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2025/52|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2025-W52&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:45, 22 December 2025 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29831856 --&gt;

== Tech News: 2026-03 ==

&lt;section begin=&quot;technews-2026-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/03|Translations]] are available.

'''Weekly highlight'''
* The Wikimedia Foundation has shared some guiding questions for the July 2026–June 2027 Annual Plan on [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2026-2027/Product &amp; Technology OKRs|Meta]] and ''[[diffblog:2025/12/10/shaping-wikimedia-foundations-2026-2027-annual-goals-key-questions-for-the-wikimedia-movement/|Diff]]''. These focus on global trends, faster and healthier experimentation, better support for newcomers, strengthening editors and advanced users, improving collaboration across projects, and growing and retaining readership. Feedback and ideas are welcome on the [[m:Talk:Wikimedia Foundation Annual Plan/2026-2027|talk page]].

'''Updates for editors'''
* As part of the current work of Community Tech team on the [[m:Special:MyLanguage/Community Wishlist/W372|Multiple watchlists]] project, the display of [[Special:EditWatchlist|EditWatchlist]] will be updated as a first step towards multiple watchlists. Additionally, the pagination on [[Special:Search|Search]] will be updated too, as a part of the work on the [[m:Special:MyLanguage/Community Wishlist/W186|Revamp pagination / page navigation]] wish. [https://phabricator.wikimedia.org/T411596]
* [[m:Special:GlobalWatchlist|The Global Watchlist]] is a MediaWiki [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] that lets you see your watchlists from different wikis on the same page. It was recently updated to look more like the regular [[Special:Watchlist|Watchlist]], such as preparing it for temporary accounts in IP masking (including rerouting user links to contributions pages), making page titles bold, and opening links in edit summaries and tags in new browser tabs. [https://phabricator.wikimedia.org/T398361][https://phabricator.wikimedia.org/T298919][https://phabricator.wikimedia.org/T273526][https://phabricator.wikimedia.org/T286309]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where global blocks did not have the option to disable sending emails, has now been fixed, and will be available for use in the week of January 13. [https://phabricator.wikimedia.org/T401293]

'''Updates for technical contributors'''
* The [[mw:Special:MyLanguage/VisualEditor/Citation tool|VisualEditor citation tool]] and [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] now support &quot;map&quot; as a reference type. [https://phabricator.wikimedia.org/T411083]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.10|MediaWiki]]/[[mw:MediaWiki 1.46/wmf.11|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:33, 12 January 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29907192 --&gt;

== Tech News: 2026-04 ==

&lt;section begin=&quot;technews-2026-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/04|Translations]] are available.

'''Updates for editors'''
* The tray shown on [[Special:Diff|Special:Diff]] in mobile view has been redesigned. It is now collapsed by default, and incorporates a link to undo the edit being viewed, making it easier for mobile editors and reviewers to take action while keeping the interface uncluttered. [https://phabricator.wikimedia.org/T402297] 
* [[m:Special:GlobalWatchlist|The Global Watchlist]] lets you view your watchlists from multiple wikis on one page. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] continues to improve — it now automatically determines the text direction (ensuring correct display of sites with unusual domain names) and shows detailed descriptions for log actions. Later this week, a new permanent link for page creations and CSS classes for each entry element will be added. [https://phabricator.wikimedia.org/T412505][https://phabricator.wikimedia.org/T287929][https://phabricator.wikimedia.org/T262768][https://phabricator.wikimedia.org/T414135]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the previously observed issue in Vector 2022, where anchor link targets were obscured by the sticky header, has now been addressed. [https://phabricator.wikimedia.org/T406114]

'''Updates for technical contributors'''
* As mentioned in the [[m:Special:MyLanguage/Tech/News/2025/44|October 2025 deprecation announcement]], MediaWiki Interfaces team will begin sunsetting all transform endpoints containing a trailing slash from the MediaWiki REST API the week of January 26. Changes are expected to roll out to all wikis on or before January 30th. All API users currently calling them are encouraged to transition to the non-trailing slash versions. Both endpoint variations can be found, compared, and tested using the [https://test.wikipedia.org/wiki/Special:RestSandbox REST Sandbox]. If you have questions or encounter any problems, please file a ticket in Phabricator to the [https://phabricator.wikimedia.org/project/view/6931/ #MW-Interfaces-Team board].
* Interactive reference documentation for the [[mw:Special:MyLanguage/Wikimedia REST API|Wikimedia REST API]] has moved. Requests to API docs previously hosted through [[mw:Special:MyLanguage/RESTBase|RESTBase]] (e.g.: &lt;code dir=ltr&gt;https://en.wikipedia.org/api/rest_v1/&lt;/code&gt;) are now redirected to the [[w:en:Special:RestSandbox|REST Sandbox]].
* The [[mw:Special:MyLanguage/Wikidata Platform|WMF Wikidata Platform team]] (WDP) has published its [[d:Special:MyLanguage/Wikidata:Wikidata Platform team/Newsletter|January 2026 newsletter]]. It includes updates on the legacy full-graph endpoint decommissioning, the User-Agent policy change, the monthly Blazegraph migration office hours, and efforts to reduce regressions caused by the legacy endpoint shutdown. As a reminder, you can [[m:Special:MyLanguage/Global message delivery/Targets/WDP team updates|subscribe to the WDP newsletter]]!
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.12|MediaWiki]]

'''Meetings and events'''
* The [[mw:Wikimedia Hackathon Northwestern Europe 2026|Wikimedia Hackathon Northwestern Europe 2026]] will take place on 13-14 March 2026 in Arnhem, the Netherlands. Applications opened mid-December and will close soon or when capacity is reached. It's a two-day, technically oriented hackathon bringing together Wikimedians from the region. Hope to see you there!

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:29, 19 January 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29943403 --&gt;

== Tech News: 2026-05 ==

&lt;section begin=&quot;technews-2026-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/05|Translations]] are available.

'''Updates for editors'''
* Wikimedia Foundation invites comments on [[m:Special:MyLanguage/Product and Technology Advisory Council/Year1 Reflections and Proposed Way Forward 2026 Update|proposed future]] of the [[:m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] until 28 February.
* All users with registered accounts can now use passkeys for [[m:Special:MyLanguage/Help:Two-factor authentication|two-factor authentication]] (2FA). Passkeys are a simple way to log in without using a second device. They verify the user's identity using a fingerprint, face scan, or a PIN code. To set up a passkey, first set up a regular 2FA method. Currently, to log in with a passkey, users must also use a password. Later this quarter, passwordless login will allow users to log in with a single click and a passkey. Users with advanced rights will also be required to have 2FA enabled. This is part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] project.
* Unregistered contributors on blocked IPs or blocked IP ranges can now interact on-wiki to appeal a block by creating a temporary account to appeal a block on the user talk page, unless the &quot;prevent this user from editing their own talk page&quot; is enabled. This solves the problem of logged-out users unable to use the default unblock process via user talk page. [https://phabricator.wikimedia.org/T398673]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the Two-Factor Authentication (2FA) methods description on the management page has been updated. It is now clearer and easier for users to understand and make use of. [https://phabricator.wikimedia.org/T332385]

'''Updates for technical contributors'''
* A new AbuseFilter variable, &lt;code&gt;account_type&lt;/code&gt;, has been added to provide a reliable way to determine the account type being created in the &lt;code&gt;createaccount&lt;/code&gt; and &lt;code&gt;autocreateaccount&lt;/code&gt; actions. As part of this change, the variable &lt;code&gt;accountname&lt;/code&gt; has been renamed to &lt;code&gt;account_name&lt;/code&gt;, and &lt;code&gt;accountname&lt;/code&gt; is now deprecated. Edit filter managers should update any filters that use hardcoded account type checks or the deprecated variable. [https://phabricator.wikimedia.org/T414049]
* Image thumbnails that are requested in non-standard sizes, and using non-standard methods such as direct requests to &lt;code dir=ltr&gt;&lt;nowiki&gt;upload.wikimedia.org/…&lt;/nowiki&gt;&lt;/code&gt; will stop working in the near future. This change is to prevent ongoing external abuse by web-scrapers and bots. Some users with custom CSS/JS, Interface Admins who can fix gadgets and local skins, and Tool-authors, will need to update their code to use standard thumbnail sizes. [[phab:T414805|Details, search-links, and examples of how to fix them, are available in the task]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.13|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:17, 26 January 2026 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=29969530 --&gt;

== Tech News: 2026-06 ==

&lt;section begin=&quot;technews-2026-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/06|Translations]] are available.

'''Updates for editors'''
* The &quot;{{int:pageinfo-toolboxlink}}&quot; feature, which gives validating information about a page ([{{fullurl:{{FULLPAGENAME}}|action=info}} example]), now automatically includes a table of contents. If there is a local [[{{ns:8}}:Pageinfo-header]] page created by individual users, it can now be removed. [https://phabricator.wikimedia.org/T363726]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, VisualEditor previously added bold or italic formatting inside link descriptions, making the wikicode complex. This has now been fixed. [https://phabricator.wikimedia.org/T409669]

'''Updates for technical contributors'''
* There was no XML dump on 20 January. Additionally, from now on, dumps will be generated once per month only. [https://phabricator.wikimedia.org/T414389]
* The MediaWiki Interfaces team removed support for all transform endpoints containing a trailing slash from the [https://www.mediawiki.org/wiki/Special:MyLanguage/API:REST%20API MediaWiki REST API]. All API users currently calling those endpoints are encouraged to transition to the non-trailing slash versions. If you have questions or encounter any problems, please file a ticket in phabricator to the [https://phabricator.wikimedia.org/project/view/6931/ #MW-Interfaces-Team board].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.14|MediaWiki]]

'''Weekly highlight'''
* Users are reminded that the Wikimedia Foundation has shared some guiding questions for the July 2026–June 2027 Annual Plan on [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2026-2027/Product &amp; Technology OKRs|Meta]] and ''[[diffblog:2025/12/10/shaping-wikimedia-foundations-2026-2027-annual-goals-key-questions-for-the-wikimedia-movement/|Diff]]''. These focus on global trends, faster and healthier experimentation, better support for newcomers, strengthening editors and advanced users, improving collaboration across projects, and growing and retaining readership. Feedback and ideas are welcome on the [[m:Talk:Wikimedia Foundation Annual Plan/2026-2027|talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:43, 2 February 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30000986 --&gt;

== Tech News: 2026-07 ==

&lt;section begin=&quot;technews-2026-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/07|Translations]] are available.

'''Updates for editors'''
* [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] Logged-in contributors who manage large or complex watchlists can now organise and filter watched pages in ways that improve their workflows with the new [[mw:Special:MyLanguage/Help:Watchlist labels|Watchlist labels]] feature. By adding custom labels (for example: pages you created, pages being monitored for vandalism, or discussion pages) users can more quickly identify what needs attention, reduce cognitive load, and respond more efficiently. This improves watchlist usability, especially for highly active editors.
* A new feature available on [[Special:Contributions|Special:Contributions]] shows [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] that are likely operated by the same person, and so makes patrolling less time-consuming. Upon checking contributions of a temporary account, users with access to temporary account IP addresses can now see a view of contributions from the related temporary accounts. The feature looks up all the IPs associated with a given temporary account within the data retention period and shows all the contributions of all temporary accounts that have used these IPs. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts#February 2026: Improvements to the patroller tooling|Learn more]]. [https://phabricator.wikimedia.org/T415674] 
* When editors preview a wikitext edit, the reminder box that they are only seeing a preview (which is shown at the top), now has a grey/neutral background instead of a yellow/warning background. This makes it easier to distinguish preview notes from actual warnings (for example, edit conflicts or problematic redirect targets), which will now be shown in separate warning or error boxes. [https://phabricator.wikimedia.org/T414742]
* The [[m:Special:GlobalWatchlist|Global Watchlist]] lets you view your watchlists from multiple wikis on one page. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] continues to improve — it now properly supports more than one Wikibase site, for example both [[d:|Wikidata]] and [[testwikidata:|testwikidata]]. In addition, issues regarding text direction have been fixed for users who prefer Wikidata or other Wikibase sites in right-to-left (RTL) languages. [https://phabricator.wikimedia.org/T415440][https://phabricator.wikimedia.org/T415458]
* The automatic &quot;magic links&quot; for ISBN, RFC, and PMID numbers have been [[mw:Special:MyLanguage/Help:Magic links|deprecated in wikitext since 2021]] due to inflexibility and difficulties with localization. Several wikis have successfully replaced RFC and PMID magic links with equivalent external links, but a template was often required to replace the functionality of the ISBN magic link. There is now a new [[mw:Special:MyLanguage/Help:Magic words#isbn|built-in parser function]] &lt;code dir=ltr&gt;&lt;nowiki&gt;{{#isbn}}&lt;/nowiki&gt;&lt;/code&gt; available to replace the basic functionality of the ISBN magic link. This makes it easier for wikis who wish to migrate off of the deprecated magic link functionality to do so. [https://phabricator.wikimedia.org/T145604]
* Two new wikis have been created: 
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35401|Jju]] ([[w:kaj:|&lt;code&gt;w:kaj:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T413283]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q1186896|Nawat]] ([[w:ppl:|&lt;code&gt;w:ppl:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T413273]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. 

'''Updates for technical contributors'''
* A new global user group has been created: [[{{int:grouppage-local-bot}}|{{int:group-local-bot}}]]. It will be used internally by the software to allow community bots to bypass rate limits that are applied to abusive [[w:en:Web scraping|web scrapers]]. Accounts that are approved as bots on at least one Wikimedia wiki will be automatically added to this group. It will not change what user permissions the bot has. [https://phabricator.wikimedia.org/T415588]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.15|MediaWiki]]

'''Meetings and events'''
* The [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Spring 2026|MediaWiki Users and Developers Conference, Spring 2026]] will be held March 25–27 in Salt Lake City, USA. This event is organized by and for the third-party MediaWiki community. You can propose sessions and register to attend. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/AZBWVI46SDEB65PGR5J6E4TYOQQEZXM7/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:30, 9 February 2026 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30026671 --&gt;

== Tech News: 2026-08 ==

&lt;section begin=&quot;technews-2026-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/08|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Wikimedia Site Reliability Engineering|SRE Team]] will be performing a cleanup of Wikimedia's [[m:Special:MyLanguage/Etherpad|Etherpad]] instance, the web-based editor for real-time collaborative document editing. All pads will be permanently deleted after 30 April, 2026 – if there are still migration projects in progress at that point the team can revisit the date on a case by case basis. Please create local backups of any content you wish to keep, as deleted data cannot be recovered. This cleanup helps reduce database size and minimize infrastructure footprint. Etherpad will continue to support real-time collaboration, but long-term storage should not be expected. Additional cleanups may occur in the future without prior notice. [https://phabricator.wikimedia.org/T415237]

'''Updates for editors'''
* The Information Retrieval team will be launching an [[mw:Special:MyLanguage/Readers/Information Retrieval/Phase 1|Android mobile app experiment]] that tests hybrid search capabilities which can handle both semantic and keyword queries. The improvement of on-platform search will enable readers to find what they’re looking for directly on Wikipedia more easily. The experiment will first be launched on Greek Wikipedia in late February, followed by English, French, and Portuguese in March. [https://diff.wikimedia.org/2026/01/08/semantic-search-making-it-easier-to-find-the-information-readers-want/ Read more] on Diff blog. [https://www.mediawiki.org/wiki/Readers/Information_Retrieval]
* The Reader Growth team will run [[mw:Special:MyLanguage/Readers/Reader Growth/WE3.10.2 Mobile Table of Contents|an experiment]] for mobile web users, that adds a table of contents and automatically expands all article sections, to learn more about navigation issues they face. The test will be available on Arabic, Chinese, English, French, Indonesian, and Vietnamese Wikipedias.
* Previously, site notices ([[{{ns:8}}:Sitenotice]] and [[{{ns:8}}:Anonnotice]]) would only render on the desktop site. Now, they will render on all platforms. Users on mobile web will now see these notices and be informed. Site administrators should be prepared to test and fix notices on mobile devices to avoid interference with articles. To opt out, interface admins can add &lt;code dir=&quot;ltr&quot;&gt;#siteNotice { display: none; }&lt;/code&gt; to [[{{ns:8}}:Minerva.css]]. [https://phabricator.wikimedia.org/T138572][https://phabricator.wikimedia.org/T416644]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue on [[Special:RecentChanges|Special:RecentChanges]] has been fixed. Previously, clicking hide in the active filters caused the &quot;view new changes since…&quot; button to disappear, though it should have remained visible. The button now behaves as expected. [https://phabricator.wikimedia.org/T406339]

'''Updates for technical contributors'''
* New documentation is now available to help editors debug on-site search features. It supports troubleshooting when pages do not appear in results, when ranking seems unexpected, and when you need to inspect what content is being indexed, helping make search behavior easier to understand and analyze. [[mw:Help:CirrusSearch/Debug|Learn more]]. [https://phabricator.wikimedia.org/T411169]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.16|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:17, 16 February 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30086330 --&gt;

== Tech News: 2026-09 ==

&lt;section begin=&quot;technews-2026-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/09|Translations]] are available.

'''Weekly highlight'''
* [[mw:Special:MyLanguage/Edit check/Reference Check|Reference Check]] has been deployed to English Wikipedia, completing its rollout across all Wikipedias. The feature prompts newcomers to add a citation before publishing new content, helping reduce common citation-related reverts and improve verifiability. In A/B testing, the impact was substantial: newcomers shown Reference Check were approximately 2.2 times more likely to include a reference on desktop and about 17.5 times more likely on mobile web. [https://analytics.wikimedia.org/published/reports/editing/reference_check_ab_test_report_final_2025.html]
'''Updates for editors'''
* The [[mw:Special:MyLanguage/Extension:InterwikiSorting|InterwikiSorting extension]], which allowed for the [[m:Special:MyLanguage/Interwiki sorting order|sorting of interwiki links]], has been undeployed from Wikipedia. As a result, editors who had enabled interwiki link sorting in non-compact mode (full list format) will now see links reordered. The links moving forward will be listed in the alphabetical order of language code. [https://phabricator.wikimedia.org/T253764]
* Later this week, people who are editing a page-section using the mobile visual editor, will notice a new &quot;Edit full page&quot; button. When tapped, you will be able to edit the entire article. This helps when the change you want to make is outside the section you initially opened. [https://phabricator.wikimedia.org/T387175][https://phabricator.wikimedia.org/T409112]
* [[mw:Special:MyLanguage/Readers/Reader Experience|The Reader Experience team]] is inviting editors to assess whether dark mode should still be considered &quot;beta&quot; on their wiki, based on their experience of how well it functions on desktop and mobile. If the feature is deemed mature, editors can update the interface messages in &lt;code dir=ltr&gt;MediaWiki:skin-theme-description&lt;/code&gt; and &lt;code dir=ltr&gt;MediaWiki:Vector-night-mode-beta-tag&lt;/code&gt; to indicate that dark mode is ready and no longer considered beta.
* The improved [[mw:Wikimedia_Apps/Team/iOS/Activity_Tab|Activity tab]] which displays user-insights is now available to all users of the Wikipedia iOS app (version 7.9.0 and later). Following earlier A/B testing that showed higher account creation among users with access to the feature, it has been rolled out to 100% of users along with some updates. The Activity tab now shows your edited articles in the timeline, offers editing impact insights like contribution counts and article view trends, and customization options to improve in-app experience for users.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug that prevented [[mw:Special:MyLanguage/Extension:DiscussionTools|DiscussionTools]] from working on mobile has now been fixed, restoring full functionality. [https://phabricator.wikimedia.org/T415303]

'''Updates for technical contributors'''
* The [[m:Special:GlobalWatchlist|Global Watchlist]] lets you view your watchlists from multiple wikis on one page. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] that makes this possible continues to improve. The latest upgrade is the inclusion of a [[mw:Extension:GlobalWatchlist#hook|new hook]], &lt;code dir=ltr&gt;ext.globalwatchlist.rebuild&lt;/code&gt;, which fires after each watchlist rebuild. This allows you to run gadgets and user scripts for the Special page. [https://phabricator.wikimedia.org/T275159]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.17|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:03, 23 February 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30119102 --&gt;

== Tech News: 2026-10 ==

&lt;section begin=&quot;technews-2026-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/10|Translations]] are available.

'''Weekly highlight'''
* Wikipedia 25 [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments|Birthday mode]] is now live on Betawi, Breton, Chinese, Czech, Dutch, English, French, Gorontalo, Indonesian, Italian, Luxembourgish, Madurese, Sicilian, Spanish, Thai, and Vietnamese Wikipedias! This limited-time campaign feature celebrates 25 years of Wikipedia with a birthday mascot, Baby Globe. When turned on, Baby Globe is shown on [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments/article configuration|~2,500 articles]], waiting to be discovered by readers. Communities can choose to turn Birthday mode on by getting consensus from their community and asking an admin to enable the feature and customize it via [[m:Special:MyLanguage/Wikipedia 25/Easter egg experiments#Community Configuration Demo|community configuration]] on the local wiki.

'''Updates for editors'''
* [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new feature to re-use references with different details has been released to Swedish Wikipedia, Polish Wikipedia and [[:phab:T418209|a couple of other wikis]]. You can [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#test|try the feature]] on these projects or on testwiki and [https://en.wikipedia.beta.wmcloud.org/wiki/Sub-referencing betawiki]. Learnings from the first pilot wiki German Wikipedia have been [[:m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing/Learnings|published in a report]]. Reach out to the Wikimedia Deutschland team if you are [[:m:Talk:WMDE Technical Wishes/Sub-referencing#Pilot wikis|interested in becoming a pilot wiki]].
* [[mw:Special:MyLanguage/Help:Edit check#Paste check|Paste Check]] will become available at all Wikipedias this week. The feature prompts newcomers who are pasting text they are not likely to have written into VisualEditor to consider whether doing so risks a copyright violation. Paste Check [[mw:Special:MyLanguage/Edit check/Tags|tags]] all edits where it is shown for potential review. Local administrators can configure various aspects of the feature via [[{{#special:EditChecks}}]]. [[mw:Special:MyLanguage/Edit check/Paste Check#A/B Experiment|Research]] across 22 wikis found that Paste Check resulted in an 18% decrease in relative reverted-edits compared to the control group. Translators can [https://translatewiki.net/w/i.php?title=Special%3ATranslate&amp;group=ext-visualeditor-ve-mw-editcheck&amp;filter=&amp;optional=1&amp;action=translate help to localize] this and related features.
* The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] will be standardizing the user menu in the top right for all mobile users so that it is closer to the desktop experience. Currently this user menu is only visible to users with Advanced Mobile Controls (AMC) turned on. The only change is that a couple buttons previously in the left-side menu will move to the top right for users who do not have AMC turned on. This change is expected to go out March 9 and seeks to improve the user interface. [https://phabricator.wikimedia.org/T413912]
* Starting in the week of March 2, the emails sent out when an email address was added, removed, or changed for an account will switch to a substantially nicer and clearer HTML email from the prior plaintext one. [https://phabricator.wikimedia.org/T410807]
* Notifications are currently limited to 2,000 historic entries per user, and extend back to 2013 when the feature was released. This is going to be changed to only store Notifications from the last 5 years, but up to 10,000 of them. This will help with long-term infrastructure health and help to prevent more recent notifications from disappearing too soon. [https://phabricator.wikimedia.org/T383948]
* The [[m:Special:GlobalWatchlist|Global Watchlist]] which lets you view your watchlists from multiple wikis on a single page continues to see improvements. The latest update improves label usage experience. The [[mw:Special:MyLanguage/Extension:GlobalWatchlist|extension]] now allows activating the [[mw:Special:MyLanguage/Manual:Language#Fallback languages|language fallback system]] for Wikidata items without labels in the viewed language, and showing those labels in the user’s preferred Wikidata language if no &lt;code dir=ltr&gt;uselang=&lt;/code&gt; URL parameter is provided. [https://phabricator.wikimedia.org/T373686][https://phabricator.wikimedia.org/T416111]
* The Wikipedia Android team has started a beta test of [[mw:Special:MyLanguage/Readers/Information Retrieval/Phase 1|hybrid search]] on Greek Wikipedia. Hybrid search capabilities can handle both semantic and keyword queries enabling readers to find what they’re looking for directly on Wikipedia more easily.
* For security reasons, members of certain user groups are [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|required to have two-factor authentication]] (2FA) enabled. Currently, 2FA is required to use the group, but not to be a member of it. Given that this model still has some vulnerabilities, the situation will [[phab:T418580|gradually change in March]]. Members of these groups will be unable to disable last 2FA method on their account, and it will be impossible to add users without 2FA to these groups. Users will still be able to add new authentication methods or remove them, as long as at least one method is continuously enabled. In the second half of March, users without 2FA will be removed from these groups. This applies to: CentralNotice administrators, checkusers, interface administrators, suppressors, Wikidata staff, Wikifunctions staff, WMF Office IT and WMF Trust &amp; Safety. Nothing will change for other users. See the linked task for deployment schedule. [https://phabricator.wikimedia.org/T418580]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue preventing users from creating an instance in [https://www.wikibase.cloud/ Wikibase.cloud] has now been fixed. [https://phabricator.wikimedia.org/T416807]

'''Updates for technical contributors'''
* To help ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]], over the next month the Wikimedia Foundation will implement global API rate limits across our APIs. In early March, stricter limits will be applied to unidentified requests from outside Toolforge/WMCS and API requests that are made from web browsers. In April, higher limits will be applied to identified traffic. These limits are intentionally set as high as possible to minimise impact on the community. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]].
* The Wikidata Query Service Linked Data Fragment (LDF) endpoint will be decommissioned in February. This endpoint served limited traffic, which was successfully migrated to other data access methods that were better suited to support existing use cases. The hardware used to support the LDF endpoint will be reallocated to support the ongoing backend migration efforts. [https://phabricator.wikimedia.org/T415696]
* The new Parsoid parser [[mw:Special:MyLanguage/Parsoid/Parser Unification/Updates|continues to be deployed to additional wikis]], improving platform sustainability and making it easier to introduce new reading and editing features. Parsoid is now the default parser on 488 WMF wikis (268 Wikipedias), now covering more than 10% of all Wikipedia page views.
* The process and criteria for [[Special:MyLanguage/Wikimedia Enterprise#Access|requesting exceptional access]] to the high volume feed of the ''Wikimedia Enterprise'' APIs (at no cost for mission-aligned usecases), [[m:Talk:Wikimedia Enterprise#Exceptional access criteria|have now been published]]. This is to provide more thorough and clearer documentation for users.
* [https://techblog.wikimedia.org/ Tech Blog], the blog dedicated to the Wikimedia technical community [https://techblog.wikimedia.org/2026/02/24/a-tech-blog-diff/ will be migrating] to [[diffblog:|Diff]], the community news and event blog. The migration should be complete in April 2026, after which new posts will be accepted for publishing. Readers will be able to access posts – old and new – on the landing page at https://diff.wikimedia.org/techblog.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.18|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:51, 2 March 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30137798 --&gt;

== Tech News: 2026-11 ==

&lt;section begin=&quot;technews-2026-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/11|Translations]] are available.

'''Weekly highlight'''
* [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 25 March 2026 at [https://zonestamp.toolforge.org/1774450800 15:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies.
* Last week, all wikis had 2 hours of read-only time, and extended unavailability for user-scripts and gadgets. This was due to a security incident which has since been resolved. Work is ongoing to prevent re-occurrences. For current information please see the [[m:Steward's noticeboard#Statement on Meta about today's user script security incident|post on the Stewards' noticeboard]] ([[m:Special:MyLanguage/Wikimedia Foundation/Product and Technology/Product Safety and Integrity/March 2026 User Script Incident|translations]]).

'''Updates for editors'''
* Users facing multiple blocks on mobile will now see the reasons for each block separately, instead of a generic message. This helps them understand why they are blocked and what steps they can take to resolve the issue. For example, users affected for using common VPNs (such as [[Special:MyLanguage/Apple iCloud Private Relay|iCloud Private Relay]]) will receive clearer guidance on what they need to do to start editing again. [https://phabricator.wikimedia.org/T357118]
* Later this week, [[mw:Special:MyLanguage/VisualEditor/Suggestion Mode|Suggestion Mode]] will become available as a beta feature within the visual editor at all Wikipedias. This feature proactively suggests various types of actions that people can consider taking to improve Wikipedia articles, and learn about related guidelines. The feature is locally configurable, and can also be locally expanded with custom Suggestions. Current settings can be seen at [[Special:EditChecks]] and there are [[mw:Special:MyLanguage/Help:Suggestion mode#For administrators %E2%80%93 local customization|instructions for how administrators can customize]] the links to point to local guidelines. The feature is connected to [[mw:Special:MyLanguage/Help:Edit check|Edit check]] which suggests improvements while someone is writing new content. In the future, the Editing team plans to evaluate the feature's impact with newcomers through a controlled experiment. [https://phabricator.wikimedia.org/T404600]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where the cursor became misaligned during the use of CodeMirror’s syntax highlighting, which makes wikitext and code easier to read, has now been fixed. This problem specifically affected users who defined a font rule in a custom stylesheet while creating a new topic with DiscussionTools. [https://phabricator.wikimedia.org/T418793]

'''Updates for technical contributors'''
* API rate limiting update: To help ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]], global API rate limits will be applied this week to requests without a compliant User-Agent that originate from outside Toolforge/WMCS and to unauthenticated requests made from web browsers. Higher limits will be applied to identified traffic in April. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]].
* The new GraphQL API has been released. The API was developed as a flexible alternative to select features of the Wikidata Query Service (WDQS), to improve developer experience and foster adaptability, and efficient data access. Try it out and [[d:Wikidata:Wikibase GraphQL#Feedback and development|give feedback]]. You can also [https://greatquestion.co/wikimediadeutschland/GraphQLAPI/apply sign up for usability tests].
* The [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group|PTAC Unsupported Tools Working Group]] continued improvements to [[commons:Special:MyLanguage/Commons:Video2commons#|Video2Commons]] in February, with fixes addressing authentication errors, large-file handling, task queue visibility, and clearer upload behavior. Work is still ongoing in some areas, including changes related to deprecated server-side uploads. Read [[m:Special:MyLanguage/Product and Technology Advisory Council/Unsupported Tools Working Group#February 2026|this update]] to learn more.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.19|MediaWiki]]

'''In depth'''
* The Article Guidance team invites experienced Wikipedia editors from selected [[mw:Special:MyLanguage/Article guidance/Pilot wikis and collaborators#Collaborators|pilot wikis]] and interested contributors from other Wikipedias to fill out this questionnaire which is available in [https://docs.google.com/forms/d/e/1FAIpQLSfmLeVWnxmsCbPoI_UF2jyRcn73WRGWCVPHzerXb4Cz97X_Ag/viewform English], [https://docs.google.com/forms/d/e/1FAIpQLSd6rzr4XXQw8r4024fE3geTPFe13M_6w7Mitj-YJi0sOlWTAw/viewform?usp=header Arabic], [https://docs.google.com/forms/d/e/1FAIpQLSdok3-RfB18lcugYTUMGkpwmqG_8p760Wv4dCXitOXOszjUDw/viewform?usp=header Bengali], [https://docs.google.com/forms/d/e/1FAIpQLSfjTfYp4jEo0akA4B1e-Nfg3QZPCudUjhJzHzzDi6AHyAaMGA/viewform?usp=header Japanese], [https://docs.google.com/forms/d/e/1FAIpQLScteVoI29Aue4xc72dekk-6RYtvmMgQxzMI900UOawrFrSTWg/viewform?usp=header Portuguese], [https://docs.google.com/forms/d/e/1FAIpQLSetdxnYwL3ub2vqA7awCg5hJZPMIYcDPaiTe12rY9h0GYnVlw/viewform?usp=header Persian], and [https://docs.google.com/forms/d/e/1FAIpQLScNvfJF-Ot-4pzA4qAN771_0QDJ4Li19YcUsaTgSKW8Nc7U_Q/viewform?usp=header Turkish]. Your answers will help the team customize guidance for less experienced editors and help them learn community policies and practices while creating an article. Learn more [[mw:Special:MyLanguage/Article guidance|on the project page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:53, 9 March 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30213008 --&gt;

== Tech News: 2026-12 ==

&lt;section begin=&quot;technews-2026-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/12|Translations]] are available.

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature, also known as [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror 6]], has been used for wikitext syntax highlighting since November 2024. It will be promoted out of beta by May 2026 in order to bring improvements and new [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Features|features]] to all editors who use the standard syntax highlighter. If you have any questions or concerns about promoting the feature out of beta, [[mw:Special:MyLanguage/Help talk:Extension:CodeMirror|please share]]. [https://phabricator.wikimedia.org/T259059]
* Some changes to local user groups are performed by stewards on Meta-Wiki and logged there only. Now, interwiki rights changes will be logged both on Meta-Wiki and the wiki of the target user to make it easier to access a full record of user's rights changes on a local wiki. Past log entries for such changes will be backfilled in the coming weeks. [https://phabricator.wikimedia.org/T6055]
* On wikis using [[m:Special:MyLanguage/Flagged Revisions|Flagged Revisions]], the number of pending changes shown on [[{{#Special:PendingChanges}}]] previously counted pages which were no longer pending review, because they have been removed from the system without being reviewed, e.g. due to being deleted, moved to a different namespace, or due to wiki configuration changes. The count will be correct now. On some wikis the number shown will be much smaller than before. There should be no change to the list of pages itself. [https://phabricator.wikimedia.org/T413016]
* Wikifunctions composition language has been rewritten, resulting in a new version of the language. This change aims to increase service stability by reducing the orchestrator's memory consumption. This rewrite also enables substantial latency reduction, code simplification, and better abstractions, which will open the door to later feature additions. Read more about [[f:Special:MyLanguage/Wikifunctions:Status updates/2026-03-11|the changes]].
* Users can now sort search results alphabetically by page title. The update gives an additional option to finding pages more easily and quickly. Previously, results could be sorted by Edit date, Creation date, or Relevance. To use the new option, open 'Advanced Search' on the search results page and select 'Alphabetically' under 'Sorting Order'. [https://phabricator.wikimedia.org/T403775]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug that prevented UploadWizard on Wikimedia Commons from importing files from Flickr has now been fixed. [https://phabricator.wikimedia.org/T419263]
'''Updates for technical contributors'''
* A new special page, [[{{#special:LintTemplateErrors}}]], has been created to list transcluded pages that are flagged as containing lint errors to help users discover them easily. The list is sorted by the number of transclusions with errors. For example: [[{{#special:LintTemplateErrors}}/night-mode-unaware-background-color]]. [https://phabricator.wikimedia.org/T170874]
* Users of the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature have been using [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] for syntax highlighting when editing JavaScript, CSS, JSON, Vue and Lua content pages, for some time now. Along with promoting CodeMirror 6 out of beta, the plan is to replace CodeEditor as the standard editor for these content models by May 2026. [[mw:Special:MyLanguage/Help talk:Extension:CodeMirror|Feedback or concerns are welcome]]. [https://phabricator.wikimedia.org/T419332]
* The [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] JavaScript modules will soon be upgraded to CodeMirror 6. Leading up to the upgrade, loading the &lt;code dir=ltr&gt;ext.CodeMirror&lt;/code&gt; or &lt;code dir=ltr&gt;ext.CodeMirror.lib&lt;/code&gt; modules from gadgets and user scripts was deprecated in July 2025. The use of the &lt;code dir=ltr&gt;ext.CodeMirror.switch&lt;/code&gt; hook was also deprecated in March 2025. Contributors can now make their scripts or gadgets compatible with CodeMirror 6. See the [[mw:Special:MyLanguage/Extension:CodeMirror#Gadgets and user scripts|migration guide]] for more information. [https://phabricator.wikimedia.org/T373720]
* The MediaWiki Interfaces team is expanding coverage of REST API module definitions to include [[mw:Special:MyLanguage/API:REST API/Extensions|extension APIs]]. REST API modules are groups of related endpoints that can be independently managed and versioned. Modules now exist for [https://phabricator.wikimedia.org/T414470 GrowthExperiments] and [https://phabricator.wikimedia.org/T419053 Wikifunctions] APIs. As we migrate extension APIs to this structure, documentation will move out of the main MediaWiki OpenAPI spec and REST Sandbox view, and will instead be accessible via module-specific options in the dropdown on the [https://test.wikipedia.org/wiki/Special:RestSandbox REST Sandbox] (i.e., [[{{#Special:RestSandbox}}]], available on all wiki projects).
* The [[mw:Special:MyLanguage/Extension:Scribunto|Scribunto]] extension provides different pieces of information about the wiki where the module is being used via the [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual|mw.site]] library. Starting last week, the library also provides a [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#mw.site.wikiId|way]] of accessing the [[mw:Special:MyLanguage/Manual:Wiki ID|wiki ID]] that can be used to facilitate cross-wiki module maintenance. [https://phabricator.wikimedia.org/T146616]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.20|MediaWiki]]

'''In depth'''
* The [[m:Special:MyLanguage/Coolest Tool Award|2026 Coolest Tool Award]] celebrating outstanding community tools, is now open for nominations! Nominate your favorite tool using the [https://wikimediafoundation.limesurvey.net/435684?lang=en nomination survey] form by 23 March 2026. For more information on privacy and data handling, please see the [[foundation:Special:MyLanguage/Legal:Coolest_Tool_Award_2026_Survey_Privacy_Statement|survey privacy statement]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:35, 16 March 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30260505 --&gt;

== Tech News: 2026-13 ==

&lt;section begin=&quot;technews-2026-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/13|Translations]] are available.

'''Weekly highlight'''
* Wikimedia site users can now log in without a password using passkeys. This is a secure method supported by fingerprint, facial recognition, or PIN. With this change, all users who opt for passwordless login will find it easier, faster, and more secure to log in to their accounts using any device. The new passkey login option currently appears as an autofill suggestion in the username field. An additional [[phab:T417120|&quot;Log in with passkey&quot; button]] will soon be available for users who have already registered a passkey. This update will improve security and user experience. The [[c:File:Passwordless_login_screencast.webm|screen recording]] demonstrates the passwordless login process step by step.
* [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 25 March 2026 at [https://zonestamp.toolforge.org/1774450800 15:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies.

'''Updates for editors'''
* Wikimedia site users can now export their notifications older than 5 years using a [[toolforge:echo-chamber|new Toolforge tool]]. This will ensure that users retain their important notifications and avoid them being lost based on the planned change to delete notifications older than 5 years, as previously announced. [https://phabricator.wikimedia.org/T383948]
* Wikipedia editors in Indonesian, Thai, Turkish, and Simple English now have access to Special:PersonalDashboard. This is an [[mw:Special:MyLanguage/Moderator Tools/Dashboard|early version of an experience]] that introduces newer editors to patrolling workflows, making it easier for them to move from making edits to participating in more advanced moderation work on their project. [https://phabricator.wikimedia.org/T402647]
* The [[Special:Block]] now has two minor interface changes. Administrators can now easily perform indefinite blocks through a dedicated radio button in the expiry section. Also, choosing an indefinite expiry provides a different set of common reasons to select from, which can be changed at: [[MediaWiki:Ipbreason-indef-dropdown]]. [https://phabricator.wikimedia.org/T401823]
* Mobile editors [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#Logged-out|at several wikis]] can now see an improved logged-out edit warning, thanks to the recent updates from the Growth team. These changes released last week are part of ongoing efforts and tests to enhance [[mw:Special:MyLanguage/Contributors/Account Creation Experiments|account creation experience on mobile]] and then increase participation. [https://phabricator.wikimedia.org/T408484]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:36}} community-submitted {{PLURAL:36|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the bug that prevented mobile web users from seeing the block information when affected by multiple blocks has been fixed. They can now see messages of all the blocks currently affecting them when they access Wikipedia.

'''Updates for technical contributors'''
* Images built using Toolforge will soon get the upgraded buildpacks version, bringing support for newer language versions and other upstream improvements and fixes. If you use Toolforge Build Service, review the recent [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EMYTA32EV2V5SQ2JIEOD2CL66YFIZEKV/ cloud-announce email] and update your build configuration as necessary to ensure your tools are compatible. [https://wikitech.wikimedia.org/w/index.php?title=Help:Toolforge/Building_container_images&amp;oldid=2392097#Buildpack_environment_upgrade_process][https://phabricator.wikimedia.org/T380127]

* The [https://api.wikimedia.org/wiki/Main_Page API Portal] documentation wiki will shut down in June 2026. API keys created on the API Portal will continue to work normally. api.wikimedia.org endpoints will be deprecated gradually starting in July 2026. Documentation on the API Portal is moving to [[mw:Wikimedia APIs|mediawiki.org]]. Learn more on the [[wikitech:API Portal/Deprecation|project page]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.21|MediaWiki]]

'''In depth'''
* [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] is considering improvements to [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names|automatically generated reference names in VisualEditor]]. Please check out the [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names#Proposed solutions|proposed solutions]] and participate in the [[m:Talk:WMDE Technical Wishes/References/VisualEditor automatic reference names#Request for comment|request for comment]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:51, 23 March 2026 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30268305 --&gt;

== Tech News: 2026-14 ==

&lt;section begin=&quot;technews-2026-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/14|Translations]] are available.

'''Weekly highlight'''
* The Beta version of [[abstract:|Abstract Wikipedia]] a new Wikimedia project which is language-independent, was launched last week. The project allows communities to build Wikipedia articles in their native language, which can be readily accessed by other users in their own languages. The wiki is powered by instructions from Wikifunctions and also based on structured content from Wikidata. [[:f:Special:MyLanguage/Wikifunctions:Status updates/2026-03-26|Read more]].

'''Updates for editors'''
* The Growth team is running an A/B test to evaluate a clearer, more user-friendly message that promotes account creation on wikis. Currently when logged-out mobile users begin editing, they see a jarring warning message that can feel abrupt and discouraging. This also presents temporary account editing as the default rather than encouraging account creation. The test is running on ten Wikipedias, including Arabic, French, Spanish and German. [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#2. Improve logged-out warning message (T415160)|Read more]].
* The Wikimedia Apps team is inviting feedback on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Future of Editing on the Mobile Apps|how editing should work on the Wikipedia mobile apps]]. The discussion focuses on improving how users access editing tools when they tap &quot;Edit&quot;. This is part of a broader effort to convert readers who develop an interest in editing, to access a more user-friendly pathway to start contributing.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:45}} community-submitted {{PLURAL:45|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where citation fetching from the large newspaper archive [https://www.newspapers.com Newspapers.com] was no longer working, due to a block in [[mw:Special:MyLanguage/Citoid|Citoid]] requests, has now been fixed. [https://phabricator.wikimedia.org/T419903]

'''Updates for technical contributors'''
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.22|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:25, 30 March 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30329462 --&gt;

== Tech News: 2026-15 ==

&lt;section begin=&quot;technews-2026-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/15|Translations]] are available.

'''Updates for editors'''

* The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents|CampaignEvents extension]] now includes a new group goal-setting feature, enabling organizers to set and track event goals such as the number of articles created and participating contributors in real time. Similarly, participants can work toward shared targets and see their collective impact as the event unfolds. The feature is now available on all Wikimedia wikis. Learn more in [[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Collaborative contributions#Goal setting|the documentation]].
* [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] The new [[mw:Special:MyLanguage/Help:Watchlist labels|watchlist labels]] feature (announced in [[m:Special:MyLanguage/Tech/News/2026/07|Tech News 2026-07]]) is now available via VisualEditor, the source editor, and the 'watchstar' (or watch link, for skins that don't have a star icon). Previously it was only possible to assign labels via [[Special:EditWatchlist|EditWatchlist]]. In all three places it is a new field following the expiry field.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where talk pages on mobile with Parsoid are unusable after empty section headers, has now been fixed. [https://phabricator.wikimedia.org/T419171]

'''Updates for technical contributors'''
* The [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|sub-referencing feature]], which lets editors add details to an existing reference without duplicating it, will be gradually rolled out to [[phab:T414094|more wikis]] later this year. Wikis using the [[mw:Special:MyLanguage/Reference Tooltips|Reference Tooltips]] gadget are encouraged to update their version (typically at [[m:MediaWiki:Gadget-ReferenceTooltips.js|MediaWiki:Gadget-ReferenceTooltips.js]] as shown [https://en.wikipedia.org/w/index.php?diff=1344408362 here]) to ensure compatibility. Other reference-related gadgets may also be affected. [https://phabricator.wikimedia.org/T416304]
* All Wikinews editions will be closed and switched to read-only mode on 4 May 2026. Content will remain accessible, but no new edits or articles can be added. This closure was approved by the Board of Trustees of the Wikimedia Foundation following extended discussions. [[m:Wikimedia Foundation Board noticeboard#Board of Trustees Approves Closure of Wikinews|Read more]].
* The [[:mw:Special:MyLanguage/API:Action API|Action API]] has had several formats for requested output. One of them, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;format=php&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, is being removed soon. Please ensure your scripts or bots use the [[mw:Special:MyLanguage/API:Data formats#Output|JSON format]]. This removal should affect very few scripts and bots. [https://phabricator.wikimedia.org/T118538]
* The [[Special:NamespaceInfo|Special:NamespaceInfo]] page now includes namespace aliases. For example &quot;WP&quot; for the &quot;Project&quot; (&quot;Wikipedia&quot;) namespace on the German Wikipedia. [https://phabricator.wikimedia.org/T381455]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.23|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:19, 6 April 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30362761 --&gt;

== Tech News: 2026-16 ==

&lt;section begin=&quot;technews-2026-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/16|Translations]] are available.

'''Weekly highlight'''
* Experienced editors are invited to [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Main_Page test] the [[mw:Special:MyLanguage/Article guidance|Article guidance]] feature, designed to help less-experienced editors create well-structured, policy-compliant Wikipedia articles. Testing instructions are [[mw:Special:MyLanguage/Article guidance/Test feature guide|available]]. Also, after reviewing [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Category:Pages_using_article_guidance the outlines], please provide feedback on the [[mw:Talk:Article guidance|project talk page]]. Based on your input, the feature will be refined and transferred to the pilot Wikipedias to translate and adapt. Check out [[c:File:Article Guidance workflow demo - April 2026.webm|the video]] explaining the feature.
'''Updates for editors'''
* On most wikis, all autoconfirmed users can now use [[Special:ChangeContentModel|Special:ChangeContentModel]] page to [[mw:Special:MyLanguage/Help:ChangeContentModel|create new pages with custom content models]], such as mass message lists, making custom page formats more accessible. Check [[Special:ListGroupRights|Special:ListGroupRights]] for the status of your wiki. [https://phabricator.wikimedia.org/T248294]
* The Growth team has launched an [[mw:Special:MyLanguage/Contributors/Account_Creation_Experiments|account creation experiment]] to evaluate whether adding an account creation button to the mobile web header increases new account registrations and encourages more mobile users to contribute to the wikis. The experiment is currently live on Hindi, Indonesian, Bengali, Thai, and Hebrew Wikipedia, and targets 10% of logged-out mobile web users.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where VisualEditor could get stuck loading on Windows devices with animations turned off, has now been fixed. [https://phabricator.wikimedia.org/T382856]

'''Updates for technical contributors'''
* Starting later this week, {{int:group-abusefilter}} who have the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]] beta feature enabled will have [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] as the editor at [[Special:AbuseFilter|Special:AbuseFilter]]. This is part of the broader effort to make the user experience more consistent across all editors. [https://phabricator.wikimedia.org/T399673][https://phabricator.wikimedia.org/T419332]
* Tools and bots that access the [[mw:Special:MyLanguage/Notifications/API|Notifications API]] (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=query&amp;meta=notifications&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;) will need to update their OAuth or BotPassword grants to also include access to private notifications. [https://phabricator.wikimedia.org/T421991]
* Due to a library upgrade, listings on category pages may be displayed out of order starting on Monday, 20th April. A migration script will be run to correct this, and will take hours to days depending on the size of the wiki (up to a week for English Wikipedia). [https://phabricator.wikimedia.org/T422544]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.24|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:19, 13 April 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30380527 --&gt;

== Tech News: 2026-17 ==

&lt;section begin=&quot;technews-2026-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/17|Translations]] are available.

'''Weekly highlight'''
* After two years of development, [[mw:Special:MyLanguage/Help:Extension:CodeMirror|{{int:codemirror-beta-feature-title}}]], also known as [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror 6]], is to be promoted out of beta on Tuesday, April 21. It brings better code and wikitext readability, reduction in typing errors, and other [[mw:Special:MyLanguage/Help:Extension:CodeMirror|benefits]] to all users of the standard syntax highlighter. A huge thank you to volunteer [https://phabricator.wikimedia.org/p/Bhsd/ Bhsd] who developed many of the new features, including [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Code folding|code folding]], [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Autocompletion|autocompletion]], and [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Linting|linting]]. [https://phabricator.wikimedia.org/T259059]
* A major update to the Wikipedia app for iOS is now rolling out, redesigning the interface to align with Apple's latest &quot;Liquid Glass&quot; visual design. [https://apps.apple.com/us/app/wikipedia/id324715238 Download the latest version] and explore the update.

'''Updates for editors'''
* [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4 Reading lists|Reading lists]] is a feature which allows readers to save articles to a list for reading later. This feature is now in beta on Arabic, French, Indonesian, Vietnamese, and Chinese Wikipedias and by default for all new accounts on all Wikipedias.
* An experiment which explores extending [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews|Page Previews to mobile web]] will be launched in the week of April 20 on Arabic, English, French, Italian, Polish, and Vietnamese Wikipedias. Page Previews are pop-ups that display a thumbnail, lead paragraph, and a link to open the full article of a blue link, thereby improving content discovery. The feature is already available on desktop and in the apps. [[m:Special:MyLanguage/List of experiments in Product and Technology#Template|Read more about this experiment and others]].
* On several wikis, logged-in editors who haven't [[mw:Special:MyLanguage/Help:Email confirmation|confirmed their email addresses]] can now see a banner encouraging them to do so. Having the email address confirmed allows a user to restore access to the account if they lose it. [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security#Encouraging users to confirm their email addresses|Learn more]]. [https://phabricator.wikimedia.org/T421366]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:15}} community-submitted {{PLURAL:15|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where editing very large wiki pages in the 2017 wikitext editor caused slow loading, preview and scrolling lag, and performance issues when selecting, cutting, or pasting content, has now been fixed. [https://phabricator.wikimedia.org/T184857]

'''Updates for technical contributors'''
* As part of the promotion of [[mw:Special:MyLanguage/Help:Extension:CodeMirror|CodeMirror]] from a beta feature, all users will use [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] instead of [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]] for syntax highlighting when editing JavaScript, CSS, JSON, Vue and Lua content pages. [https://phabricator.wikimedia.org/T419332]
* The &lt;code&gt;mirrors.wikimedia.org&lt;/code&gt; service for Debian and Ubuntu users will sunset and stop working on May 15. The resources for the service will be replaced with new and better options. Some users may need to switch to a different server which should take about a minute. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LJYRIS4WB66HIRCAO4GIDTXCMDVZRBMA/ You can read more]. [https://phabricator.wikimedia.org/T416707]
* The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;image&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;oldimage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; table will be removed from [[wikitech:Help:Wiki Replicas|wikireplicas]]. If your tools or queries access &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;image&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;oldimage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; directly, please update them to use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;file&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;filerevision&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; table before 28 May. [https://phabricator.wikimedia.org/T28741]
* Following the recent implementation of global API rate limits on unidentified traffic, the Wikimedia Foundation will continue efforts to ensure [[mw:Special:MyLanguage/MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]] by applying global limits to identified API traffic beginning the last week of April. These limits are intentionally set as high as possible to minimise impact on the community. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, see [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]] and [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits/FAQ|Frequently Asked Questions]].
* The [[mw:Special:MyLanguage/Attribution API|Attribution API]] is now available as a [[mw:Special:MyLanguage/Wikimedia APIs/Stability policy|beta]]. The API fetches information for crediting Wikimedia articles and media files wherever they are used. Reference documentation is available through the REST Sandbox special page available on all Wikimedia wikis (such as the [https://en.wikipedia.org/w/index.php?api=attribution.v0-beta&amp;title=Special%3ARestSandbox REST sandbox on English Wikipedia]). Share your feedback on the [[mw:Talk:Attribution API|project talk page]].
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:00, 20 April 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30432763 --&gt;

== Tech News: 2026-18 ==

&lt;section begin=&quot;technews-2026-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/18|Translations]] are available.

'''Updates for editors'''
* There is a change in how new users are autoconfirmed that will improve anti-vandalism protection. Currently, users who have had an account for a few days and made a few edits are automatically added to the [[{{int:grouppage-autoconfirmed/{{CONTENTLANGUAGE}}}}|{{int:group-autoconfirmed}}]] group. This configuration tends to be exploited by some vandals, who create accounts and start to use them only after some time. To mitigate this, the configuration will be updated next week so that – for the purpose of becoming autoconfirmed – the account age will be counted from their first edit, instead of registration date. The numeric value of the age threshold will remain the same. This change will be deployed only to wikis which require at least one edit as part of the autoconfirmation conditions. [https://phabricator.wikimedia.org/T418484]
* All Wikipedia users with new accounts and those who activated the &quot;automatically enable most beta features&quot; option in their preference can now use the [[mw:Special:MyLanguage/Readers/Reader Experience/WE3.3.4 Reading lists|reading lists]] beta feature to save articles for later reading. This helps organize reading interests in one place for convenient access. 
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the issue where infobox images have huge padding in Firefox, has been fixed. [https://phabricator.wikimedia.org/T423676]

'''Updates for technical contributors'''
* As a reminder, the global API rate limits will be applied this week to identified API traffic. This is to help ensure [[mw:MediaWiki Product Insights/Responsible Reuse|fair use of infrastructure]]. Bots running in Toolforge/WMCS or with the bot user right on any wiki should not be affected for now. However, all developers are advised to follow updated best practices. For more information, including the actual rate limits, see [[mw:Wikimedia APIs/Rate limits|Wikimedia APIs/Rate limits]] and [[mw:Wikimedia APIs/Rate limits/FAQ|Frequently Asked Questions]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.26|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:06, 27 April 2026 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30458046 --&gt;

== Tech News: 2026-19 ==

&lt;section begin=&quot;technews-2026-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/19|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Article guidance|Article guidance]] team invites experienced editors of [[mw:Special:MyLanguage/Article guidance/Pilot wikis and collaborators|pilot Wikipedias]]—Arabic, Bangla, Japanese, Portuguese, Persian, Turkish, Simple English, Spanish, and French—to help translate and adapt [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Category:Pages_using_article_guidance sample outlines]. These outlines will guide editors in creating clear, well-structured, and policy-compliant articles when using [https://b24e11a4f1.catalyst.wmcloud.org/wiki/Special:NewArticle the feature] once it is launched in May 2026. [[mw:Special:MyLanguage/Article guidance#Adapting a sample outline in a Wikipedia|Simple instructions]] on how to translate and adapt the outlines are available.
'''Updates for editors'''
* The [[:m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] has published [[:m:Special:MyLanguage/Product and Technology Advisory Council/May 2026 draft PTAC recommendation for feedback|draft recommendations]] on a model that affiliates can follow when contributing to the technical space. Community members are invited to provide feedback on the recommendation until May 8th [[:m:Talk:Product and Technology Advisory Council/May 2026 draft PTAC recommendation for feedback|on the talk page]].
* The number of available thumbnail size preferences in MediaWiki is being reduced to three standardized options—Small (180px), Regular (250px), and Large (400px), as part of ongoing efforts to improve performance and reduce strain on thumbnail services. As a result, existing preferences will be mapped to the nearest new size (for example, smaller selections like 120px or 150px will render at 180px, while larger ones like 300px or 360px will render at 400px). The preferences interface will soon be updated to reflect these changes, and users who wish to opt out or provide feedback can do so. [https://phabricator.wikimedia.org/T424909]
* From now on, even when a permission expires automatically, users will receive an Echo notification similar to the standard notification for permission changes. There is a difference between this and [[m:Special:MyLanguage/Global reminder bot|Global reminder bot]] in that the latter reminds users a week ''before'' the rights are due to expire, so that they can renew the rights.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the problem where the ULS language selector in [[m:Special:Translate|Special:Translate]] would scroll vertically when it shouldn't, has been resolved. Previously, when users opened the &quot;Translate to English&quot; dropdown and typed certain inputs, the dialog would scroll vertically by a few pixels even when there was enough space to display all results. The dropdown no longer shifts unnecessarily when filtering languages. [https://phabricator.wikimedia.org/T358864] 
* The [[m:Special:GlobalWatchlist|Global Watchlist]], which lets you view your watchlists from multiple wikis on a single page, continues to improve. For example, watchlists for Wikibase sites such as [[:d:|Wikidata]] now support [[mw:Special:MyLanguage/Extension:EntitySchema|EntitySchema]] elements for better tracking. The Live Updates mode now refreshes the special page every 60 seconds to comply with the updated [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|global API rate limits]] for improved real-time responsiveness. Additionally, a directionality bug that displayed links as &quot;changes 3&quot; instead of &quot;3 changes&quot; in mixed-direction lists has been fixed. [https://phabricator.wikimedia.org/T415450][https://phabricator.wikimedia.org/T424422][https://phabricator.wikimedia.org/T418091]
'''Updates for technical contributors'''
* The second phase of [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits|global API rate limits]] has been rolled out to reduce the [[diffblog:2026/03/26/quo-vadis-crawlers-progress-and-whats-next-on-safeguarding-our-infrastructure/|impact of AI crawlers]] and ensure fair, sustainable access to Wikimedia resources, prioritising human and mission-aligned traffic. [[mw:Special:MyLanguage/Wikimedia APIs/Rate limits#Limits|Limits]] have been shifted from per-hour to per-minute, producing smoother traffic patterns and more predictable API load. Community users are not expected to be affected, and no action is required. Early indications show some User-Agent-based requestors are adjusting behaviour, and around 64% of automated API traffic has been identified. Monitoring continues, and Wikimedia Enterprise remains available for commercial support.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.46/wmf.27|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:43, 4 May 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30498077 --&gt;

== Tech News: 2026-20 ==

&lt;section begin=&quot;technews-2026-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/20|Translations]] are available.

'''Weekly highlight'''
* Community Tech has published [[m:Special:MyLanguage/Community Wishlist/How to write a good wish|new guidance]] explaining how wishes on Community Wishlist are triaged and prioritized. The documentation is intended to help contributors write stronger proposals by clarifying the factors that influence prioritization decisions. Beyond vote counts, the guidance highlights considerations such as potential impact on the community when determining which wishes move forward.

'''Updates for editors'''
* The Reader Growth team is launching an experiment to test a new [[mw:Special:MyLanguage/Readers/Reader_Growth/Share_Card|Share Card feature]] that allows readers to create visually engaging cards from Wikipedia articles or selected article sections and share them online, with each card linking back to the original article to help expand readership and article discovery. The mobile-only A/B test will be available to a portion of readers on Arabic, Chinese, French, Vietnamese, and English Wikipedia to better understand reading and sharing habits, and is scheduled to begin the week of May 18 and run for four weeks.
* The Android and iOS Wikipedia apps recently released the [[mw:Special:MyLanguage/Wikimedia_Apps/Team/25th_Birthday_Reading_Challenge|25-day reading challenge]] into Beta, as part of efforts to drive reader engagement by encouraging users to complete reading milestones. To track their reading streak during the challenge, App users can add a widget featuring Baby Globe to their home screen. The challenge officially begins May 11.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:17}} community-submitted {{PLURAL:17|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the global preference for enabling syntax highlighting in wikitext could unexpectedly disable itself after being turned on, has now been fixed. [https://phabricator.wikimedia.org/T425286]

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The ResourceLoader module &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mediawiki.ui.input&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, deprecated since [[m:Special:MyLanguage/Tech/News/2023/39|September 2023]], will be removed this week. There is a [[mw:Special:MyLanguage/Codex/Migrating_from_MediaWiki_UI|guide for migrating from MediaWiki UI to Codex]] for any tools that use it. [https://phabricator.wikimedia.org/T420125]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.2|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:20, 11 May 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30524429 --&gt;

== Tech News: 2026-21 ==

&lt;section begin=&quot;technews-2026-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/21|Translations]] are available.

'''Weekly highlight'''
* The Abstract Wikipedia team has identified five potential pilot wikis to assess their interest in adopting abstract articles on their wikis. The pilots are Malayalam, Bengali, Dagbani, Arabic, and Indonesian Wikipedia. The feedback period will be open until May 22. If your community is interested in becoming a pilot, [[m:Talk:Abstract Wikipedia|let us know on Meta]].

'''Updates for editors'''
* An experiment to show [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists]] to logged-out readers on mobile web will launch on May 18 across German, Spanish, Italian, Portuguese, Polish, Dutch, Turkish, and Urdu Wikipedias, and will run for one month. The effort supports broader goals of helping readers save and organize articles for later reading, while encouraging habits that could lead to future Wikipedia contributions.
* To support a bookmark button in the Reading List beta feature, the &quot;Tools &gt; Action&quot; menu has been updated to display icons, including the watch star indicator that helps editors identify temporarily watched articles. The icons now also match those used on mobile, improving consistency across platforms. The change is currently limited to the actions menu and mainly affects editors with privileged user rights. [https://phabricator.wikimedia.org/T426008]
* [[mw:Special:MyLanguage/VisualEditor/Suggestion Mode|Suggestion Mode]] was released as an [[w:en:A/B test|A/B test]] for newcomer editors on the mobile website at [[phab:T421189|~15 Wikipedias]]. The experiment will measure the impact that Suggestion Mode has on the proportion of newcomer mobile web edit sessions that result in constructive (un-reverted) article edits. The experiment will also evaluate the feature's impact on editor retention, and monitor changes in revert and block rates.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue in the Wikipedia Android app where images could sometimes fail to load after opening a recommended reading list notification, has now been fixed. [https://phabricator.wikimedia.org/T418231]

'''Updates for technical contributors'''
* The [[mw:Special:MyLanguage/Wikidata Platform|Wikidata Platform team]] has published its [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/Backend Replacement|backend replacement recommendation]] and accompanying [[wikitech:Wikidata Query Service/WDQS Architecture re-design|technical architecture]] for the migration of the Wikidata Query Service (WDQS) away from Blazegraph. Feedback is invited until May 25th 2026, especially on potential gaps and impacts on advanced use cases. Wikidata community members and WDQS users are also encouraged to help identify high-impact tools and workflows that may need attention on [[d:Wikidata:SPARQL query service/WDQS backend update/High-Impact Use Cases|this page]]. Feedback can be shared on the [[d:Wikidata talk:SPARQL query service/WDQS backend update|Migration talk page]] or during the [[d:Special:MyLanguage/Wikidata:Blazegraph Migration Office Hours|next office hour]]. See the [[d:Special:MyLanguage/Wikidata:Wikidata Platform team/Newsletter|WDP team newsletter]] for more details.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.3|MediaWiki]]

'''In depth'''

* On English, French, Japanese, and a few other Wikipedias, there was a [[diffblog:2025/09/02/better-detecting-bots-and-replacing-our-captcha/|trial of hCaptcha]], a third-party bot detection service. The trial showed that hCaptcha effectively detects and deters some bad-faith automated activity, on its own and by giving [[w:en:Wikipedia:Village pump (technical)/Archive 225#Introducing SuggestedInvestigations|checkusers and stewards]] signals to look into. Because the results were positive, hCaptcha will be rolled out across all wikis over the next few weeks. [[mw:Special:MyLanguage/Product Safety and Integrity/Anti-abuse signals/hCaptcha|See the hCaptcha project page]] for technical information about the implementation and privacy protections. [[diffblog:2026/05/04/better-detecting-bots-and-replacing-our-captcha-part-2/|Learn more]].
* The latest Community Tech update is now available, with progress across several Community Wishlist initiatives, including Reading Lists expansion from the mobile app to the website, new language support for &quot;Who Wrote That&quot; and the Personal Dashboard, improvements to 3D rendering and Charts, and upcoming work on talk page sorting, audio playback, and editing workflows. The update also shares current priorities, wishlist status trends, and opportunities for community feedback on future focus areas and the Wikimedia Foundation’s 2026–2027 Annual Plan. [[m:Special:MyLanguage/Community Wishlist/Updates#May 13, 2026: Latest updates from the Community Tech team|Read the full newsletter for details]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W21&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:21, 18 May 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30539262 --&gt;

== Tech News: 2026-22 ==

&lt;section begin=&quot;technews-2026-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/22|Translations]] are available.

'''Weekly highlight'''
* Following a [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#LOWM|successful account creation experiment]], an improved logged-out edit warning message will be deployed to all Wikimedia wikis in the first week of June. The change will only affect logged-out users on mobile web who open an editing session. The updated experience is designed to encourage account creation more clearly, while still allowing users to edit with temporary accounts. Results from the experiment showed a significant increase in account creation, with a 27% relative lift among users shown the updated message. As expected, as more people funnel into account creation, temporary accounts decreased by a relative 16%. The experiment did not show any significant changes in constructive edit rates or other monitored contributor metrics. [https://phabricator.wikimedia.org/T424595] 

'''Updates for editors'''
* For security reasons, members of certain user groups are [[m:Special:MyLanguage/Mandatory two-factor authentication for users with some extended rights|required to have two-factor authentication]] (2FA) enabled. Members of these groups will be unable to disable the last 2FA method on their account, and it will be impossible to add users without 2FA to these groups. Users will still be able to add new authentication methods or remove them, as long as at least one method is continuously enabled. In the next few weeks, users without 2FA will be removed from these groups. Notably, this applies to bureaucrats. See the linked tasks for deployment schedules. [https://phabricator.wikimedia.org/T423119][https://phabricator.wikimedia.org/T423120]
* [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] will run an [[w:en:A/B testing|A/B test]] on [[:phab:T415904|10 wikis]], testing [[m:WMDE Technical Wishes/References/Reference Previews|potential improvements for Reference Previews]]. The experiment will run for ~2 weeks at the end of May / beginning of June and will affect 10% of desktop readers on the participating wikis.
* After two successful experiments, the Reader Growth team is rolling out an [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|Image Browsing]] beta feature for all Wikipedias on mobile on May 25. This means that anyone who has all beta features on by default will start to see this feature, and others can check the box to turn it on in their preferences. The beta feature will include a carousel of all an article's images at the top of the article, with controls for editors to [[mw:Readers/Reader_Growth/Image_Browsing#Phase_2.1_beta_feature|exclude images from the article's carousel or to exclude an article from the feature entirely]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, three dimensional STL files were being rendered incorrectly by the media viewer 3D extension which is now fixed. [https://phabricator.wikimedia.org/T416723]

'''Updates for technical contributors'''
* The legacy CSS classes &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tleft&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tright&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; have been replaced with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;floatleft&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;floatright&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; as the former do not work consistently across all MediaWiki platforms, notably mobile web and mobile apps. Projects relying on these classes are encouraged to review related usage and plan for migration. Please note that &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;floatleft&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;floatright&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; may also be deprecated in future, although there are currently no plans to do so. [[phab:T426452|Read more]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.4|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:52, 25 May 2026 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30584502 --&gt;

== Tech News: 2026-23 ==

&lt;section begin=&quot;technews-2026-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/23|Translations]] are available.

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] is conducting an experiment to show the [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|reading lists]] feature, which is still in development, to logged-out mobile readers to test whether it encourages account creation at a higher rate compared to the watchstar button. The [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists#Experiment timeline|experiment]] was launched on May 18th on German, Spanish, Italian, Portuguese, Polish, Dutch, Turkish, and Urdu wikis, and it will run for a month.
* The Wikimedia Apps team released [[mw:Special:MyLanguage/Wikimedia Apps/Team/Explore Feed Refresh/Phase 1|Phase 1]] of the redesigned Home Feed to the Android Beta app. The new Home Feed includes a refreshed &quot;Community&quot; tab and a personalized &quot;For You&quot; tab featuring daily updated reading recommendations. The redesign is part of a broader effort to improve content discovery and create more engaging learning experiences in the Wikipedia apps.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:18}} community-submitted {{PLURAL:18|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where images could fail to load for some suggested edits on [[w:Special:Homepage|Special:Homepage]], leaving the thumbnail stuck in a loading state, has now been fixed. [https://phabricator.wikimedia.org/T424048]

'''Updates for technical contributors'''
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.5|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:08, 1 June 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30613639 --&gt;

== Tech News: 2026-24 ==

&lt;section begin=&quot;technews-2026-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/24|Translations]] are available.

'''Weekly highlight'''
* Wikimedia Enterprise has increased the free usage limits for its API offerings. The monthly request limit for the On-demand API has increased from 5,000 to 50,000 requests, while the Snapshot API limit has increased from 15 to 30 requests per month. In addition, Structured Contents snapshots are now available for free accounts. These changes expand access to Wikimedia Enterprise data for developers, researchers, and organizations using Wikimedia content. [https://enterprise.wikimedia.com/blog/enhanced-free-api]

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Explore Feed Refresh/Phase 1|refreshed Explore Feed]], now called the Home Feed, is rolling out to 50% of users of the Wikipedia Android app. The Home Feed helps readers discover relevant content through two new tabs: ''Community'' and ''For You''. The Community tab provides a scrollable feed of curated content and updates from the broader Wikimedia community and movement, while the ''For You'' tab offers a full-screen, swipeable experience that shows content tailored to a user's interests. The redesign is part of a broader effort to improve discovery and enhance the learning experience in the Wikipedia app.
* The [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/&quot;Which came first?&quot; Game|Which came first?]] daily trivia game is now available in the beta version of the Wikipedia iOS app in English, German, French, Portuguese, Russian, Spanish, Arabic, Chinese, and Turkish. The game uses historical events from Wikipedia's &quot;On This Day&quot; content and challenges readers to guess which of two events happened first. The game was previously released on Android. Communities interested in making the game available in their languages can [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Games#Game availability by language|read the instructions and requirements]].
* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]], a new MediaWiki feature that allows editors to reuse references with different details, will begin rolling out to Wikimedia wikis following a successful pilot phase. Deployment will start on 8 June for most [[wikitech:Deployments/Train#Wednesday|Group 1 wikis]] and French Wikipedia, with additional Wikipedia language editions receiving the feature over the coming months. Communities are encouraged to prepare by checking for [https://translatewiki.net/w/i.php?title=Special%3ATranslate&amp;group=ext-cite&amp;language=en&amp;action_source=search&amp;filter=%21translated&amp;optional=1&amp;action=translate untranslated Cite extension messages] in their language and reviewing any use of [[mw:Special:MyLanguage/Reference Tooltips|Reference Tooltips]], which may require [[:phab:T416304#11668731|updates]] to support the new functionality. Wikis using [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] do not need to take any action. Communities may also wish to create the ''cite-tracking-category-ref-details'' [[Special:TrackingCategories|tracking category]] as a hidden category using &lt;code&gt;&lt;nowiki&gt;__HIDDENCAT__&lt;/nowiki&gt;&lt;/code&gt; (or a dedicated template), and connect it to the corresponding Wikidata item [[d:Q129764848]]. [https://phabricator.wikimedia.org/T425662]
* The [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews#Experimentation|Page Previews experiment]] on mobile web has concluded. The team decided not to roll out the feature after the results showed no statistically significant impact on reader retention, as the primary success metric was retention improvement. Page Previews, which are already available on desktop and in the apps, display a thumbnail, lead paragraph, and link to the full article when readers tap a blue link. The experiment tested this experience on mobile web across six Wikipedias.
* The [[mw:Special:MyLanguage/Codex/Design/Icons|user interface icon library]] will be [[phab:T399175|updated later this week or next week]]. Most of the ~300 icons have been slightly refined and ~30 new icons have been added. These changes improve the icons to make them more consistent and comprehensible, and provide more visual balance when they are used in groups.
* The [[mw:Special:MyLanguage/Universal Language Selector|Universal Language Selector]] (ULS) interface in MediaWiki, which helps users select content in other languages, has been updated. The new version improves speed and accessibility, and users of Wikimedia projects can now pin languages for quicker language switching. The deployment to Wikimedia sites will happen gradually in the coming weeks. You can test it now as a beta feature by selecting [[Special:Preferences#mw-prefsection-betafeatures|beta features]] in your profile preferences and share your feedback on [[mw:Special:MyLanguage/Universal Language Selector/New ULS|the project page]]. 
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:21}} community-submitted {{PLURAL:21|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the Pageviews Analysis dashboard on pageviews.wmcloud.org stopped updating graph data in May 2026, affecting all users, has been fixed. [https://phabricator.wikimedia.org/T427171]

'''Updates for technical contributors'''
* The function signature for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mw.util.addPortletLink()&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; has been simplified. Developers can now pass a configuration object instead of a list of positional parameters when creating portlet links. The previous function signature remains supported for backwards compatibility. For example, instead of: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mw.util.addPortletLink('p-cactions', '#', 'Stub', 'ca-stubtag', 'Add a stub tag to this page');&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mw.util.addPortletLink('p-cactions', { href: '#', text: 'Stub', id: 'ca-stubtag', tooltip: 'Add a stub tag to this page' });&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Script maintainers are encouraged to review existing uses of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;addPortletLink()&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and update them where appropriate. This change will be available on all wikis from 11 June. Thanks to community volunteer Gerges for contributing this improvement. [https://phabricator.wikimedia.org/T427945]
* '''Community Wishlist discussion''': Product &amp; Technology [[m:Special:MyLanguage/Community Wishlist/Updates#May 20, 2026: Community Tech becomes a program|introduced changes]] meant to increase the number and complexity of wishes fulfilled, including the disbanding of the Community Tech team. They are [[m:Special:MyLanguage/Community Wishlist/Updates|engaging in discussions]] about a [[m:Talk:Community Wishlist#Proposed direction for Wishlist|proposed direction for the wishlist]] from community members. Includes ways to structure annual voting, better tracking of wishes, removing focus areas, and [[m:Special:MyLanguage/Community Wishlist/Updates|staffing updates]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.6|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:30, 8 June 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30650573 --&gt;

== Tech News: 2026-25 ==

&lt;section begin=&quot;technews-2026-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/25|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Readers/Reader Growth|Reader Growth team]] has launched an [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|Image Browsing]] beta feature on the mobile web version of all Wikipedias. The feature shows an image carousel at the top of articles with 3 or more images. Editors can configure this feature with the following controls: to hide a specific image from a page, either use &lt;code&gt;class=notpageimage&lt;/code&gt; excluding it from thumbnail previews, or &lt;code&gt;class=noviewer&lt;/code&gt; excluding it from MediaViewer. The carousel can also be disabled from a page entirely, with the magic word &lt;code&gt;&lt;nowiki&gt;__NOMEDIAVIEWERCAROUSEL__&lt;/nowiki&gt;&lt;/code&gt;. To submit feedback or flag bugs, please visit the [[mw:Talk:Readers/Reader Growth/Image Browsing|project page]]. 
* [[mw:Special:MyLanguage/Help:Tables#class=&quot;wikitable&quot;|Wikitables]] can now be [[mw:Special:MyLanguage/Help:Sortable tables#Forcing the initial sort direction|sorted in descending order]] on the first click by adding &lt;code dir=ltr&gt;data-sort-order=&quot;desc&quot;&lt;/code&gt; to the header cell. Previously, by default, clicking a column header for the first time sorts it in ascending order. This addition to a Wikitable gives it more control and flexibility, while the default behavior for subsequent clicks remains unchanged. [https://phabricator.wikimedia.org/T398416]

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Article guidance|Article guidance]] feature is currently being tested with some editors creating new articles on the Simple English, French, and Turkish Wikipedias. The experiment will soon begin on the Arabic and Bangla Wikipedias as well. [[w:simple:Special:NewArticle|This feature]] gives editors community-curated guidance to help them create articles that follow community standards. Experienced editors can continue creating or adapting outlines for specific article types that are commonly created by less experienced contributors. The outlines guide less experienced editors in creating high-quality articles. A quick guide to markups used in outlines can be found on [[mw:Special:MyLanguage/Article guidance/Test feature guide#Markups in outlines|this page]]. [[w:simple:Wikipedia:Article Guidance|Example outlines]] that can be adapted and instructions for how to adapt them are on [[mw:Special:MyLanguage/Article guidance#Adapting a sample outline in a Wikipedia|this section]] of the project page.
* Wikis that wish to replace the &quot;indefinitely&quot; button in Special:Block for temporary accounts (for example, wikis that block temporary users only until account expiration) will be able to do so by creating [[MediaWiki:ipb-indefinite-expiry-temporary-account]] with the block duration they want. [https://phabricator.wikimedia.org/T427125]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:41}} community-submitted {{PLURAL:41|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* By the end of June, a valid user-agent string will be required for automated dumps downloads from the dumps.wikimedia.org website. Automated requests that provide a generic or empty user-agent will be blocked. This [[phab:T400119|extends enforcement]] of the long standing [[foundation:Special:MyLanguage/Policy:Wikimedia Foundation User-Agent Policy|user-agent policy]]. Access to dumps through Wikimedia Cloud Services will not change.
* The roll out of global [[mw:Wikimedia APIs/Rate limits|API rate limits]] is now complete, with limits enforced across all APIs and at the documented levels for all groups. Bots running in Toolforge/WMCS or with the bot user right on any wiki remain exempt. All bots should continue to follow the documented best practices to avoid being rate limited.
* The [https://api.wikimedia.org/wiki/Main_Page API Portal wiki] will be read only starting this week (June 15-18). The following week (June 22-25), all API Portal wiki URLs will redirect to [[mw:Wikimedia APIs|Wikimedia APIs on mediawiki.org]]. Learn more on the [[wikitech:API Portal/Deprecation|project page]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.7|MediaWiki]]

'''Meetings and events'''
* On June 17th at 6pm UTC the WMF will be holding Discord call focused on a code review. We've heard through the [[mw:Special:MyLanguage/Developer Satisfaction Survey/2026|Developer Satisfaction Survey]] that volunteers are struggling with code review and we'd like to discuss these experiences with the goal of surfacing workable solutions. You can join the call [https://discord.gg/wikipedia?event=1514727511102062664 via the Wikimedia Community Discord server].
* The [[m:Special:MyLanguage/Conferencia Wikimedia de América Latina 2026|Latin American Wikimedia Conference]] will host a regional hackathon that will bring together the Wikimedia movement’s technical community including developers, system administrators, data scientists, and users with extended rights. Interested technical contributors can [https://docs.google.com/forms/d/e/1FAIpQLSf4osJzTHBJjQbYJk7TMVEJjTEQv7IgtsUDfP-o-qTgeRQQxw/viewform apply for a scholarship] to participate until June 21 at midnight (Bolivia time, UTC-4).
* Sign up for Wikimania Team Challenges to join this special event. The Team challenges will take place online and in person from July 21 to 22, before Wikimania conference. Everyone is welcome, regardless of skills or Wikimania registration. Teams will work on 10 important challenges supporting the Wikimedia community. For details, visit [[wmania:Special:MyLanguage/2026:Team challenges|the Team Challenges page]] and [https://wikimedia.eventyay.com/wm/teamchallenges/ register there]. Registration closes on June 20th at 11pm UTC. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:48, 15 June 2026 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30689604 --&gt;

== Tech News: 2026-26 ==

&lt;section begin=&quot;technews-2026-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/26|Translations]] are available.

'''Weekly highlight'''
* [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]] are [[phab:T418115|now available at Wikidata]]. This update enables access to Mentorship ([[mw:Special:MyLanguage/Help:Growth/Mentorship|if configured]]), Impact module, the Help Panel, and a simplified Newcomer Homepage (without Suggested Edits). Wikidata administrators are still configuring the features through Community Configuration.

'''Updates for editors'''
* The special page [[{{#special:RangeCalculator}}]] has been created. It allows users to find an IP range without needing to rely on external tools. Until now, this tool was only available to CheckUsers. [https://phabricator.wikimedia.org/T268429]
* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|Sub-referencing]] is a new MediaWiki feature that allows editors to reuse references with different details. It will be deployed to most small and medium-sized Wikipedia language versions on June 23. The [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#deployment|FAQ]] lists possible actions to take on your wiki to support the deployment. Check the [[:phab:T414094|rollout plan]] for the next deployment steps. [https://phabricator.wikimedia.org/T428902]
* Starting next week, users will get a notification when they are blocked or unblocked from editing, or if this block changes. [https://phabricator.wikimedia.org/T100974]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Starting next week, abuse filters that are set to &quot;require CAPTCHA verification&quot; will begin to also affect users with the &lt;code&gt;skipcaptcha&lt;/code&gt; right, which includes most autoconfirmed users. Bots are exempted. This change only affects edits that trigger an abuse filter. The &lt;code&gt;skipcaptcha&lt;/code&gt; right will continue to exempt users from having to solve CAPTCHAs in the ordinary course of using the wikis. [https://phabricator.wikimedia.org/T402595]
* Reference documentation for the [[wikitech:Machine_Learning/LiftWing/API|Lift Wing API]] has moved from the API Portal to the interactive [https://wikitech.wikimedia.org/w/index.php?api=lift-wing&amp;title=Special%3ARestSandbox REST Sandbox].
* The API Portal wiki is now closed. For API documentation, see [[mw:Special:MyLanguage/Wikimedia_APIs|Wikimedia APIs on mediawiki.org]]. All API Portal wiki URLs (https://api.wikimedia.org/wiki/) will redirect to the mediawiki.org page starting June 22. [https://phabricator.wikimedia.org/T427537]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.8|MediaWiki]]

'''Meetings and events'''
* Join an online call on 25 June at 2:30pm UTC to meet the current Wikimedia interns for [[mw:Google_Summer_of_Code/2026|Google Summer of Code]] and [[mw:Outreachy/Round_32|Outreachy]]. Interns will provide an overview of their projects and a brief demo of their work so far. Attendees are encouraged to [[mw:event:Google_Summer_of_Code/Summer_2026_June_Internship_open_session|share ideas and connections in their community]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 13:05, 23 June 2026 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30722494 --&gt;

== Tech News: 2026-27 ==

&lt;section begin=&quot;technews-2026-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/27|Translations]] are available.

'''Updates for editors'''
* As part of the [[mw:Special:MyLanguage/Contributors/Account Creation Experiments|Account Creation Experiments]], the Growth team tested adding a user account icon in the mobile web header for logged-out users, providing direct access to &quot;Create account&quot; and &quot;Log in&quot; actions. The experiment increased account creation by about 20% without negatively affecting edit quality or constructive edit rates. The feature will now be rolled out to all Wikimedia Foundation wikis on mobile web in the first week of July. [https://phabricator.wikimedia.org/T428220]
* After a [[phab:T426248|successful experiment]], logged-in users who did not [[mw:Special:MyLanguage/Help:Email_confirmation|confirm their email address]] when their account was created see a new banner asking them to complete that process. This helps reduce the risk that users get locked out of their account, and makes account email addresses overall more reliable. This is part of the [[mw:Special:MyLanguage/Product Safety and Integrity/Account Security|Account Security]] project. [https://phabricator.wikimedia.org/T428292]
* An update to [[Special:Search|Search]] is refining how the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;-prefix:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; behaves when used to exclude results. Previously, using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;-prefix:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; with negation could unintentionally broaden search results by adding the namespaces included in the search scope, leading to confusing behavior for users expecting a straightforward exclusion filter. With the update, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;-prefix:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will now strictly exclude matching page titles as intended and may display a warning if the relevant namespace has not been explicitly selected. The behavior of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;prefix:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; without negation however remains unchanged. [https://phabricator.wikimedia.org/T427443]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:33}} community-submitted {{PLURAL:33|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where reviewers using the Page Curation toolbar were not automatically subscribed to talk page discussions they started has now been fixed. Reviewers will now receive notifications when someone replies to those discussions. [https://phabricator.wikimedia.org/T329346]

'''Updates for technical contributors'''
* Starting June 29th, automated downloads from the dumps.wikimedia.org website will be subject to the [[Foundation:Special:MyLanguage/Policy:Wikimedia Foundation User-Agent Policy|user-agent policy]]. Automated requests that provide a generic or empty user-agent will be blocked. Access to dumps through Wikimedia Cloud Services remains unaffected. This is a follow up to the announcement made in the [[m:Special:MyLanguage/Tech/News/2026/25|2026/25 issue of Tech News]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.9|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 11:48, 29 June 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30744833 --&gt;

== Tech News: 2026-28 ==

&lt;section begin=&quot;technews-2026-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/28|Translations]] are available.

'''Updates for editors'''
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:34}} community-submitted {{PLURAL:34|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the search bar results on Wikidata, showed English results instead of using the correct language fallback for users of language variants, has now been fixed. Search suggestions will now follow the expected language fallback chain. [https://phabricator.wikimedia.org/T429769]

'''Updates for technical contributors'''
* In preparation for [[m:Special:MyLanguage/Event:Celebrate Women|Celebrate Women campaign]] planned for March 2027, the Wikimedia Foundation’s [[m:Special:MyLanguage/Wikimedia Foundation/Advancement/Community Growth/Content Enablement|Content Enablement team]] has launched a 22-question survey to better understand technical contributions by women+ (anyone who identifies as a woman) across Wikimedia projects. The survey takes approximately 15–20 minutes to complete and will remain open until 20 July 2026. The [[m:Special:MyLanguage/Celebrate Women/Technical contributions survey|questions]] are also available on-wiki for review in advance.
* The [[mw:Special:MyLanguage/Extension:Score|Score extension]] now supports rendering music scores as SVG images in addition to PNG, addressing a long-standing [[:phab:T49578|feature request]] and resolving historical image quality issues. Both formats are now provided to clients, with PNG in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;src&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute and SVG in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;srcset&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute.
* The new [[wikitech:Parsoid|Parsoid]] parser [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Updates|continues to be deployed to additional wikis]], making it easier to introduce new reading and editing features. It was enabled on French Wikipedia, bringing total progress to covering 78.9% of Wikipedia page views. Rollout to English Wikipedia desktop will progress through this week.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.10|MediaWiki]]

'''In depth'''
* The Wikimedia Hackathon 2026 [[diffblog:2026/06/29/wikimedia-hackathon-2026-building-collaborating-and-shaping-the-future-together/|recap blog post]] is now live. It highlights the projects, sessions, and social activities from this year’s event, and shares initial plans for the 2027 Wikimedia Hackathon.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 13:57, 6 July 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30773578 --&gt;

== Tech News: 2026-29 ==

&lt;section begin=&quot;technews-2026-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/29|Translations]] are available.

'''Updates for editors'''
* [[mw:Special:MyLanguage/Growth/Revise_Tone|Revise Tone]] helps newcomers identify passages in Wikipedia articles that may contain non-encyclopedic language and encourages them to consider revising the tone. The feature was [[w:en:A/B_testing|A/B tested]] on the Arabic, English, French, and Portuguese Wikipedias, where newcomer task completion rates [[mw:Special:MyLanguage/Growth/Revise_Tone#Experiment_Results|increased by 38.7%]] compared to the default Copyedit task, with no decrease in edit quality. The test ended on July 9, and the feature is now available for everyone on these wikis, configurable via Community Configuration. [[phab:T426364|The plan]] is to release Revise Tone to more wikis.
* The community configuration that allows [[mw:Special:MyLanguage/Help:Growth/Mentorship#Automated mentor list cleanup|automatic removal of inactive mentors]] based on configurable criteria will be enabled on Thursday 16, [[mw:Special:MyLanguage/Growth/Deployment|on some wikis]] to keep mentor lists up to date. Mentors are experienced contributors who opt in to help new users on-wiki through the [[mw:Special:MyLanguage/Growth/Feature summary|Growth Features]]. Administrators can now prepare the settings via [[w:Special:CommunityConfiguration/Mentorship|Special:CommunityConfiguration/Mentorship]]; they will take effect starting Thursday.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:38}} community-submitted {{PLURAL:38|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where some users of the Wikipedia Android app were logged out immediately after signing in, preventing them from staying logged in and editing pages, has now been fixed. [https://phabricator.wikimedia.org/T316916]

'''Updates for technical contributors'''
* Editing a page via user scripts or gadgets was causing watchlist labels that the user had assigned to that page to reset. This has now been fixed. [https://phabricator.wikimedia.org/T423778]
* To work around a Safari bug (see [[phab:T425211]]), on Parsoid-enabled wikis, wikilink hrefs now use absolute urls instead of protocol-relative urls. REST API output remains unchanged and continue to use protocol-relative urls. Gadgets, user scripts, bots, and CSS might need to be adapted if they relied on the presence of protocol-relative urls in wikilink hrefs. [https://phabricator.wikimedia.org/T431358]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.11|MediaWiki]]

'''In depth'''
* The Wikimedia Foundation’s Experiment Platform Team has published a blog post reflecting on its first year of structured experimentation. It highlights successful experiments such as Paste Check, Reference Check, and Tone Check, which improved editing outcomes and have been rolled out to more users, as well as experiments that did not lead to product changes. [[diffblog:2026/07/07/moving-the-needle-how-we-test-new-ideas-across-wikimedia-projects|Read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:11, 13 July 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30804401 --&gt;

== Tech News: 2026-31 ==

&lt;section begin=&quot;technews-2026-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/31|Translations]] are available.

'''Updates for editors'''
* [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] [[mw:Special:MyLanguage/ContentTranslation|Content Translation]] now supports dark mode, fulfilling a [[m:Community Wishlist/W544|Community Wishlist request]]. This brings the tool in line with the accessibility features available in the Vector 2022 and Minerva skins, helping reduce visual fatigue for users translating content. [https://phabricator.wikimedia.org/T367077]
* DiscussionTools' source mode and the 2017 wikitext editor will now offer autocomplete for links (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;[[&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), templates (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), HTML and parser tags (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), and magic words (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;), making it quicker and easier to insert links, templates, and other wiki markup while editing. [https://phabricator.wikimedia.org/T432400]
* The [[mw:Special:MyLanguage/Readers/Reader Growth/Mobile page previews|Readers Growth team]] has concluded its experiment with mobile page previews and will not roll out the feature. Page Previews are a pop-up bottom sheet that appears when readers tap a blue link, showing a thumbnail, lead paragraph, and an option to open the article. The experiment showed flat retention and negative indicator metrics, suggesting that mobile web readers preferred navigating directly to linked articles rather than using page previews.
* The Reader Experience team has seen encouraging early results from the [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists feature]], with 93% of participating users reporting that it was useful. Reading Lists help active readers save articles for future reading and support their learning goals on Wikimedia projects. The team plans further improvements before expanding the feature to more users.
* The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Explore Feed Refresh|Explore Feed Refresh]] initiative was tested with new and casual Wikipedia app readers. The refreshed feed helps readers discover new and relevant content. After a 10.5% increase in engagement with the feed, Wikimedia Apps team has decided to scale the Home Feed redesign to iOS with the learnings from the Android release applied.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where subject names in the Article Guidance feature were displayed with incorrect capitalization on French Wikipedia, has now been fixed. Subject names will now follow the correct capitalization rules for the language. [https://phabricator.wikimedia.org/T427201]

'''Updates for technical contributors'''
* After running several [[mw:Special:MyLanguage/Contributors/Account Creation Experiments|Account Creation Experiments]] to improve registration completion rates, a new version of the username field on [[Special:CreateAccount|Create Account]] has been rolled out. It includes [[:c:File:Create account - July 2026 updates.png|a popover summarizing the username policy]] to provide clearer guidance during account creation. As part of this change, the messages &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-helpusername&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-help&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; that several communities have configured will no longer be used. If communities want to customize the guidance shown in the new popover, they can instead edit the following messages: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-policy-popover-bullet1&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-policy-popover-bullet2&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;createacct-username-policy-popover-bullet3&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T430604]
* Later this week, the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|CodeMirror syntax highlighter]] will offer [[w:en:Theme (computing)|themes]]. The themes can be picked from a dropdown menu in the full [[mw:Special:MyLanguage/Help:Extension:CodeMirror#CodeMirror preferences|CodeMirror preferences]] dialog. For wikitext, available themes are default, colorblind-friendly (previously the colorblind preference option on [[Special:Preferences#mw-prefsection-editing]]) and no-highlighting. For code languages (i.e., CSS/JavaScript/JSON/Vue/Lua), there are several themes available. These same themes will eventually be available for wikitext, too. [https://phabricator.wikimedia.org/T163533]
* From now on, wikis can restrict editing in the &quot;User&quot; namespace to only the page owner and certain user groups. [[mw:Special:MyLanguage/Manual:$wgRestrictUserPageEditing|Read the configuration documentation]] to learn more.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.14|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:49, 27 July 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30856308 --&gt;

== Tech News: 2026-32 ==

&lt;section begin=&quot;technews-2026-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/32|Translations]] are available.

'''Updates for editors'''
* The [[mw:Special:MyLanguage/Readers/Reader Experience|Reader Experience team]] has developed a [https://82db7c8d4b.catalyst.wmcloud.org/w/index.php?title=Regent%27s_Park&amp;uselang=de patch demo] that wraps the page toolbar onto two lines when there is not enough horizontal space for all the buttons. This aims to reduce crowding in the Vector 2022 toolbar, which can occur on some language Wikipedias at certain screen widths. [https://phabricator.wikimedia.org/T429518]
* The Reader Experience team is planning to launch [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists]], a [[m:Special:MyLanguage/Community Wishlist Survey 2021/Mobile and apps/Have Apps reading lists available on Destop/Mobile|Community Wishlist item]], which is currently available to try in beta, as a full feature in September. Before then, volunteer translator help is needed for [https://translatewiki.net/w/i.php?title=Special%3ATranslate&amp;group=ext-readinglists&amp;filter=&amp;action=translate string translations] into a number of languages. The feature supports reading and learning goals on Wikipedia.
* Next week, the table of contents on Wikimedia Commons file pages will be improved by consolidating the file page table of contents with the page table of contents. This will make it easier to understand a file page’s structure, navigate to specific sections, and share links to individual sections. [https://phabricator.wikimedia.org/T332644]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:24}} community-submitted {{PLURAL:24|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where some [[w:TIFF|TIFF]] images failed to load after clicking their thumbnail, causing a broken image to be displayed instead of the full-size image, has now been fixed. [https://phabricator.wikimedia.org/T429326]

'''Updates for technical contributors'''
* The variable and function selector in AbuseFilter has been updated to support search and autocomplete. It will allow filter maintainers to find the desired variable or function more quickly. [https://phabricator.wikimedia.org/T323698]
* The MJPEG and VP8 formats are removed from the video player. The MP4 format (MPEG-4 Part 2) is added instead, which provides higher quality videos to older iPhone devices. It may take a few weeks to retroactively update all existing videos. The default format for modern devices stays the same (VP9/WebM). [https://phabricator.wikimedia.org/T358266]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.15|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:46, 3 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30872536 --&gt;

== Tech News: 2026-33 ==

&lt;section begin=&quot;technews-2026-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/33|Translations]] are available.

'''Updates for editors'''
* [[File:Maki-gift-15.svg|12px|link=|class=skin-invert|Wishlist item]] A new ChartWizard is [[c:Special:ChartWizard/Data:Example.Pie.chart|now available on Wikimedia Commons]] for users interested in creating charts from their own data. The wizard makes the [[mw:Special:MyLanguage/Extension:Chart|Chart extension]] more beginner-friendly by allowing editors to create charts, such as bar and pie charts, without needing to use JSON. Users can still switch to the JSON editor if they prefer. Feedback on the new tool is welcome on the [[m:Talk:Community Wishlist/W414|wish talk page]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the Wikipedia iOS app’s Picture of the Day widget displayed the same image every day instead of updating daily, has now been fixed. [https://phabricator.wikimedia.org/T430692]

'''Updates for technical contributors'''
* [[mw:Special:MyLanguage/Extension:Math|Math formula]] SVG images will soon be generated in the browser instead of on the server. MathML continues to be generated on the server and renders in the browser without JavaScript. Wikibooks will see this change on 12 August, Wikisource on 19 August and Wikipedia from 20-27 August. You can try this by selecting &quot;{{int:Mw-math-mathjax}}&quot; in your preferences. This change is part of [[mw:Special:MyLanguage/RESTBase/deprecation|deprecating RESTBase]] and [[phab:T431372|deprecating Mathoid]]. [https://phabricator.wikimedia.org/T271001]
* Category pages will soon support sorting entries by the time they are added to a category. This will make it easier to find recently or long-standing categorized pages. It will also improve workflows for maintenance categories such as deletion backlogs and other time-based review tasks. You can use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;cldsort=timestamp&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; URL argument in category view to sort the entries. [https://phabricator.wikimedia.org/T433768]
* [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] and user scripts on Wikimedia wikis may now use [[phab:T395347|ES2018 features]] and [[phab:T419142|ES2019 features]] in JavaScript code. Previously, the platform only allowed up to ES2017. MediaWiki validates the source code to protect functionality from syntax errors and to ensure scripts are valid in all [[mw:Special:MyLanguage/Compatibility#Browser_support_matrix|supported browsers]]. [https://phabricator.wikimedia.org/T419142]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.16|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:45, 10 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30901051 --&gt;

== Tech News: 2026-34 ==

&lt;section begin=&quot;technews-2026-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/34|Translations]] are available.

'''Weekly highlight'''
* The [[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Worklist|Worklist feature]] for the Event Registration tool is now live on all Wikimedia wikis. With Worklist, event organizers can add the articles their event will focus on directly to the event page. The Worklist also powers [[mw:Special:MyLanguage/Help:Extension:CampaignEvents/Registration/Worklist#How Event Pathways uses the worklist|Event Pathways]] which notifies other editors of the upcoming or ongoing event when they edit an article featured in the event's Worklist. This is the minimum viable version (MVP), and feedback is welcome. Organizers are encouraged to try the feature. A hands-on [[m:Special:MyLanguage/Event:Worklist Setup Workshop: Get Your Event Ready|Worklist Setup Workshop]] will take place on 18 August at 16:00 UTC and 19 August at 11:00 UTC.

'''Updates for editors'''
* [[Special:ShortPages]] displays short pages by their size, but in many cases it gets filled with disambiguations and soft redirects, making it harder to find the short articles themselves. Starting this weekend, you will be able to choose not to include an article in the special page by adding the magic word &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__EXPECTSHORTPAGE__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T433203]
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia/en}} in [[d:Q3436680|Bole]] ([[w:bol:|&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;w:bol:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;]]) [https://phabricator.wikimedia.org/T429921]
* Starting the week of August 17, the page toolbar will wrap onto two lines when there is not enough horizontal space for all the buttons. This is a [[phab:T429518|fully merged patch from the Reader Experience team]] which aims to reduce crowding in the Vector 2022 toolbar, that may occur on some language Wikipedias at certain screen widths.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:16}} community-submitted {{PLURAL:16|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, uploading large files to Wikimedia Commons has become more stable and less prone to failure following some fixes related to the “Could not acquire lock” upload error. [https://phabricator.wikimedia.org/T386640]

'''Updates for technical contributors'''
* Debian Bullseye will reach the end of its Long Term Support on 31 August 2026. [[phab:T434103|Some Cloud VPS projects]] still have instances running Debian Bullseye. Maintainers of those projects are encouraged to migrate to Debian Bookworm or Debian Trixie. A [[wikitech:Help:Cloud VPS instance operating system migration|migration guide]] is available to help with the process, and users may also want to consider whether their workload is better suited to Toolforge. If you need help or cannot complete the migration by 31 August, please contact the Cloud VPS admins as soon as possible. [[listarchive:list/cloud-announce@lists.wikimedia.org/thread/RVIPQSYLKMSL5M46JP6NEJVGVE6I2RXQ/|Read more]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.16|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W34&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:03, 17 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30922806 --&gt;

== Tech News: 2026-35 ==

&lt;section begin=&quot;technews-2026-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/35|Translations]] are available.

'''Updates for editors'''
* The [[Special:CreateAccount|Special:CreateAccount]] page has been simplified as part of ongoing work to modernize the account creation experience. The panel showing project statistics no longer appears next to the form on desktop and mobile web. Multiple account creation experiments show that a simpler form helps newcomers complete registration. [https://phabricator.wikimedia.org/T433783]
* In order to improve page performance, images now load when they are viewed. This means images lower down an article will not load if a reader never scrolls to that part of the page, which may affect some image-related metrics. [https://phabricator.wikimedia.org/T148047]

* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:42}} community-submitted {{PLURAL:42|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where image thumbnails in Abstract Wikipedia could fail to display after the corresponding file was moved on Wikimedia Commons, has now been fixed. Thumbnails will now update correctly when files are moved. [https://phabricator.wikimedia.org/T433448]

'''Updates for technical contributors'''
* User Info card is a feature that helps patrollers see information about user accounts. So far, it has been available only in places such as page history, logs and recent changes. Now, it's possible to [[mw:Special:MyLanguage/Help:Extension:CheckUser#User_Info_card_in_page_content|place it in the page content]] as well, using the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{#uic:}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; parser function. It can be particularly useful in templates like [[:en:Template:Userlinks|&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{Userlinks}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;]] (or their specialized variants), as it will make it easier to see the context about a user on various noticeboard pages. The card will be displayed only to users who have it enabled in their [[Special:Preferences#mw-input-wpcheckuser-userinfocard-enable|preferences]]. [https://phabricator.wikimedia.org/T424466]
* Due to user security and privacy risks, we have disabled access to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Special:MyPage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; URLs when specifically using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=raw&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. If you are impacted by this, consider whether you can use an alternative approach. &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Special:MyPage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; URLs can still be accessed and used without &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=raw&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Specified user page URLs (e.g. &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;User:Myusername&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;) can still be used with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=raw&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T120386]
* Due to an update, the thumbnailing software has been improved. This includes upgrading &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;librsvg&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; to 2.60 and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;ImageMagick&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; to 7, as well as resolving a number of long-standing thumbnailing bugs like rendering errors. [https://phabricator.wikimedia.org/T419815#12222841]

* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.17|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:45, 24 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30961208 --&gt;

== Tech News: 2026-36 ==

&lt;section begin=&quot;technews-2026-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/36|Translations]] are available.

'''Weekly highlight'''
* A new format for the Community Wishlist is open for feedback. You can [[m:Special:MyLanguage/Community Wishlist/Community Wishlist 2027|read the proposed ideas on Meta]]. This new process plans to improve how wishes are triaged, voted on, and prioritized in a way that is transparent and balanced across project families and language editions. This consultation is open for two weeks.

'''Updates for editors'''
* The latest release of the Wikipedia Android app includes updates to the Saved feature, bringing the app’s saving experience closer to iOS and Web. The update redesigns the Saved tab with an “All articles” view, removes the default “Saved” reading list, renames reading lists to “Collections,” and modernizes the article-saving experience. [https://phabricator.wikimedia.org/T420788]
* The [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists]] feature was enabled for all logged-in users on Bengali, Chinese, Czech and Vietnamese Wikipedias on August 25, after several months as a beta feature. Reading Lists will be available to all logged-in users on Arabic, French and Indonesian Wikipedias on September 1, followed by English Wikipedia on September 14, and all other Wikipedia wikis on September 28.
* At the end of the month, some logged-out readers on Bengali, Czech, Persian, English, and Polish Wikipedias using the Minerva skin on mobile will see an [[mw:Special:MyLanguage/Readers/Reader_Growth/Minimal_Minerva|updated navigation bar]] in an [[w:A/B test|A/B test]]. The test will compare the current navigation bar with a new version designed to make it easier to find information more quickly. The goal is to determine whether these changes encourage readers to return more often. This experiment will not change the experience for logged-in readers and/or editors.
* Editors who maintain redirects, templates, and categories used on redirect pages now have improved ways for finding and curating redirects. Previously, redirects pages could not be searched. Two new search keywords, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;onlyredirects:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;withredirects:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, now allow redirects to be searched directly and can be combined with existing keywords such as &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;incategory:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;intitle:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;insource:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T204089]
* The ISBN lookup tools for generating citations were recently not working because of external service problems. Developers are working on solutions. [https://phabricator.wikimedia.org/T435179]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:30}} community-submitted {{PLURAL:30|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where searching for pages by category using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;deepcat&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; could return no results or unrelated results has now been fixed. [https://phabricator.wikimedia.org/T414859]

'''Updates for technical contributors'''
* The domain of URLs for thumbnails is changing from upload.wikimedia.org to thumb.wikimedia.org. The old URLs will continue to work for the foreseeable future but MediaWiki will advertise the new domain instead. URLs to other types of media such as original files, videos and transcodes will still be served from upload.wikimedia.org. [https://phabricator.wikimedia.org/T427465]
* The Wikimedia [https://www.mediawiki.org/w/index.php?api=wmf-math%2Fv1&amp;title=Special%3ARestSandbox Math API] is now deprecated. These endpoints will be fully sunset by the end of September 2026. Developers who call these endpoints should transition to alternative math rendering solutions, such as the native [https://developer.mozilla.org/en-US/docs/Web/MathML MathML] or [https://www.mathjax.org/ MathJax]. Third-party MediaWiki installations that utilize the Math extension for formula rendering are required to upgrade to v1.43+ to avoid disruption of service.

* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.18|MediaWiki]]

'''In depth'''
* Read more about [[mw:Special:MyLanguage/Edit_check/TextMatch|TextMatch]] in a Diff post titled, [[diffblog:2026/08/28/custom-edit-suggestions-for-every-wiki-how-communities-are-shaping-suggestion-mode-with-textmatch/|Custom edit suggestions for every wiki: How communities are shaping Suggestion Mode with TextMatch]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:53, 31 August 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=30990659 --&gt;

== Tech News: 2026-37 ==

&lt;section begin=&quot;technews-2026-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/37|Translations]] are available.

'''Updates for editors'''
* [[mw:Special:MyLanguage/Help:Growth/Tools/Add_a_link|Add a Link]] has been upgraded for English Wikipedia users with improved detection of links that have different capitalization. This will help reduce ambiguous suggestions caused by differences in capitalization in article titles, including articles about cultural goods. A second phase of improvements is planned, which will also prepare the feature for release in other languages. [https://phabricator.wikimedia.org/T435526#12240876] [https://phabricator.wikimedia.org/T434259]
* The [[mw:Special:MyLanguage/Article_guidance|Article guidance]] feature will be enabled by default for junior editors on Simple English and Turkish Wikipedia starting 10 September 2026 following [[mw:Special:MyLanguage/Article_guidance/Updates#Summary_of_experiment_result|an experiment]]. Junior editors with 0 to 99 edits will automatically see the feature when they click a red link or use the &quot;[[tr:Vikipedi:Madde_sihirbazı|Madde oluşturto]]&quot; option on Turkish Wikipedia to create a new article. The change is intended to help junior editors create higher-quality articles that meet each Wikipedia’s standards. [[phab:maniphest/query/z3tcTjmLMxk5/#R|Additional improvements]] will continue based on the experiment results and community feedback.
* The [https://pageviews.wmcloud.org Pageviews Analysis] tool which allows users to compare pageviews across multiple pages, turns 10 years old this year and several new features have been added. They include [[toolforge:wikinav|WikiNav]] which provides insights into how readers of Wikipedia explore the content, editing stats in [https://pageviews.wmcloud.org/siteviews?range=latest-30&amp;sites=en.wikipedia.org Siteviews], the ability to [https://pageviews.wmcloud.org/massviews?source=wikiproject&amp;project=en.wikipedia.org lookup articles belonging to a WikiProject], and support for dark mode. [https://phabricator.wikimedia.org/T378549]
* A [[mw:Special:MyLanguage/Readers/Reader_Growth/Minimal_Minerva|visually simplified Minerva]] navigation bar is being tested for logged-out readers using mobile web on the Bengali, Czech, English, Farsi, and Polish Wikipedias. The experiment aims to determine whether simplifying the navigation improves reader retention. The test will run from August 31 to September 28, and no action is required from users.
* [[m:Special:MyLanguage/WMDE Technical Wishes|WMDE Technical Wishes]] is working on improving [[en:Wikipedia:VisualEditor/Named references|auto-generated reference names in VisualEditor]]. Editors will only notice a slight change starting this week. When adding automatic reference names the numbering will start at &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;:1&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; instead of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;:0&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Read more on the [[m:WMDE Technical Wishes/References/VisualEditor automatic reference names|project page]]. [https://gerrit.wikimedia.org/r/c/VisualEditor/VisualEditor/+/1332704]
*  All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only for a few minutes]] on September 23. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T433363]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where [[mw:Special:MyLanguage/Parsoid|Parsoid]] could mishandle nested nowiki tags, causing content to be lost and displaying unwanted text has now been fixed. [https://phabricator.wikimedia.org/T435116]
'''Updates for technical contributors'''
* Interface administrators can configure gadgets from [[MediaWiki:Gadgets-definition]]. The definition format has been updated and no longer requires the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;ResourceLoader&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; option, as gadgets are always loaded through &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;ResourceLoader&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This simplifies gadget configuration by removing an option that is no longer necessary, making gadget definitions easier for administrators. [https://phabricator.wikimedia.org/T298199]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.19|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:44, 7 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31008437 --&gt;

== Tech News: 2026-38 ==
&lt;section begin=&quot;technews-2026-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/38|Translations]] are available.

'''Weekly highlight'''
* The Future Audiences team has started a [[en:Special:MyLanguage/Wikipedia:Village pump (proposals)#New proposed experiment to let readers know about “Google preferred sources”|discussion on English Wikipedia]] about a proposed new experiment to show a temporary notice about the new [https://blog.google/products-and-platforms/products/search/original-high-quality-content-search/ Google Preferred Sources] feature to Wikipedia readers coming from Google. The experiment is to test whether this notice makes visitors come back to Wikipedia more often. Preferred Sources feature lets users choose websites they trust so Google can highlight content from those sources more prominently in Search and AI experiences. We are interested in testing this on other languages that Google supports and would welcome assistance with starting conversations on other wikis. If interested, please reach out to Future Audiences on the [[m:Special:MyLanguage/Future Audiences/Preferred sources|talk page]].

'''Updates for editors'''
* The chat platform Discord is releasing a new self-service framework that websites can use to specify how their links should look when shared on Discord. The Future Audiences team is planning to start using this new framework to improve how Wikipedia links look when shared on Discord, giving better attribution and credit to contributors. The link appearance and behavior won't change in the first phase of this project as we want to first collect some baseline data to assess the impact of future changes. However, if community members who are active on Discord spot any issues, they can reach out on [[phab:tag/future-audiences/|Phabricator]] or [[m:Special:MyLanguage/Future Audiences|Metawiki]].
* The Reader Growth team will be running an experiment to test whether [[mw:Special:MyLanguage/Readers/Reader_Growth/Compact Lead|compacting article lead sections on mobiles]] with the addition of a &quot;read more&quot; button improves reader retention. The test will begin on September 17 on Arabic, Spanish, French, Indonesian, Italian, Japanese, Portuguese, Vietnamese, and Chinese Wikipedias and will run for four weeks.
* All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only for a few minutes]] on September 23. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming week. [https://phabricator.wikimedia.org/T433363]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where [https://wikistats.wmcloud.org/ Wikistats] for Wikipedias was returning an HTTP 500 error and could not be reached, has now been fixed. [https://phabricator.wikimedia.org/T435959]

'''Updates for technical contributors'''
* Developers who maintain a tool that queries the Wikimedia Commons links tables need to update their code to connect to the new x4 database cluster. The links tables have been moved from the s4 cluster to x4, and will no longer receive updates on s4. The page and redirect tables remain available on both clusters. A wiki replica for the x4 cluster will be set up afterwards. The change is being made because the s4 cluster has grown too large to operate efficiently. You can [[wikitech:Special:MyLanguage/News/2026 Commons links tables database split|read more]].
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.20|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:31, 14 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31039826 --&gt;

== Tech News: 2026-39 ==
&lt;section begin=&quot;technews-2026-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/39|Translations]] are available.

'''Weekly highlight'''
* [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on Wednesday, 23 September 2026 at [https://zonestamp.toolforge.org/1790172000 14:00 UTC]. This is for the datacenter server switchover backup tests, [[wikitech:Special:MyLanguage/Deployments/Yearly calendar|which happen twice a year]]. During the switchover, all Wikimedia website traffic is shifted from one primary data center to the backup data center to test availability and prevent service disruption even in emergencies. [https://phabricator.wikimedia.org/T433363]

'''Updates for editors'''
* The Growth team tested a new post-edit notice designed to [[mw:Special:MyLanguage/Contributors/Account Creation Experiments#4. Encourage Temporary Accounts to Register|encourage Temporary Account holders to register for a permanent account]]. The new notice replaced multiple dialogs and a simultaneous welcome notification with a single message highlighting the benefits of creating an account. The experiment increased permanent account creation from 1.94% to 3.66%, an 87% relative increase. The change will now be released to all wikis. [https://phabricator.wikimedia.org/T433551]
* For Wikipedia editors using the [[Special:Preferences#mw-prefsection-betafeatures|&quot;Suggestion mode&quot; Beta Feature]], there is a [[Special:Preferences#mw-input-wpvisualeditor-editcheck-experimental|new opt-in user preference]] for showing &quot;experimental&quot; edit checks and suggestions. These types are listed at [[Special:EditChecks#experimental-checks|Special:EditChecks]], and are intended for early developer-testing and for gathering feedback from experienced users. Administrators can change the [[mw:Special:MyLanguage/Edit check/Configuration|configuration details]] for each suggestion as usual. Administrators can also [[mw:Special:MyLanguage/Help:Suggestion mode#Experimental Suggestions|use this feature]] to help test their community's ideas for locally created checks and suggestions. Experimental types will have a more distinct visual style later this week. [[mw:Talk:VisualEditor/Suggestion Mode|Feedback is welcome]].
* The [[m:CEE Technical Village Pump|CEE Technical Village Pump]] has been launched as a space for technical discussions and collaboration among Wikimedia communities in Central and Eastern Europe.
* A new tool, [[mw:Special:MyLanguage/Language Onboarding and Development/Starter kit|Starter Kit]], is now available for new and small language Wikipedia communities. It brings together guided tasks, tools, and resources to help communities get started, monitor their progress, and collaborate with the broader Wikimedia community. Starter Kit is hosted on Wikimedia Toolforge and is designed for Wikipedias with fewer than 50,000 articles. Learn more about how Starter Kit works and how communities have been using it in [[diffblog:2026/09/18/introducing-starter-kit-for-new-and-small-language-wikipedias/|this Diff blog post]].
* The Reader Growth team is launching a retest of the [[mw:Special:MyLanguage/Readers/Reader Growth/Image Browsing|image carousel experiment]]. The retest will use three new versions of the design, updated based on community feedback. The team will assess results and determine with communities whether or not to proceed with the feature. The experiment will begin the week of September 28 and will run on Arabic, Bengali, Chinese, Czech, English, Farsi, French, German, Indonesian, Japanese, Polish, Portuguese, Spanish, Swedish, and Vietnamese Wikipedias.
* After successful rollouts to Arabic, Bengali, Chinese, Czech, English, French, Indonesian, and Vietnamese Wikipedias, the [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|Reading Lists feature]] from the Reader Experience team will be available to all logged-in users on all Wikipedia wikis starting September 28.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:28}} community-submitted {{PLURAL:28|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, an issue where the mentor status script repeatedly updated mentors’ status even when nothing had changed, creating unnecessary Recent Changes entries, has now been fixed. [https://phabricator.wikimedia.org/T436659]

'''Updates for technical contributors'''
* The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;abusefilters&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; list query API, used to retrieve specific details about active or historical abuse filters configured on a wiki, has been updated to support the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;formatversion=2&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; configuration parameter, thereby leading to breaking changes to the API response. All users who maintain user-scripts or other code need to check if they use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;abusefilters&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; list query API, and respond to any breaking changes. [https://phabricator.wikimedia.org/T435828]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.21|MediaWiki]]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 11:56, 27 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31085309 --&gt;

== Tech News: 2026-40 ==
&lt;section begin=&quot;technews-2026-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2026/40|Translations]] are available.

'''Updates for editors'''
* Users of default skins on desktop and mobile web (Vector 2022 and Minerva respectively) will now be informed if the search suggestions displayed as they type into Search come from page redirects. Previously, users could get confused when the top search suggestions did not match exactly what they had typed. The new redirect notice clarifies that these suggestions are not random, but related to the search. The suggested pages may be redirects to the most relevant destination articles being searched for. [https://phabricator.wikimedia.org/T303013]
* Logged-out Wikipedia users on mobile web will begin to see a bookmark button for saving articles to their [[mw:Special:MyLanguage/Readers/Reader Experience/Reading lists|reading lists]] instead of the current Watchstar button. When clicked, the bookmark button prompts users to log in or create an account to save an article for later. An earlier experiment found that the bookmark icon was four times more likely than the Watchstar button to encourage logged-out users to create an account. This change is part of efforts to encourage more readers to become account holders and use features that help them save and return to content. [https://phabricator.wikimedia.org/T438769]
* An experiment will begin on September 29 on Spanish, Arabic, English, and French Wikipedias to simplify the experience users encounter immediately after account creation. The test will explore whether removing friction from the current Welcome Survey helps new users understand what to do next more easily. The experiment will run through the end of October. [https://phabricator.wikimedia.org/T430058]

* The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Technology Advisory Council]] has put out a call for the community to suggest topics that it should consider and discuss with the Wikimedia Foundation. Editors and technical contributors are invited to add topics to [[m:Talk:Product and Technology Advisory Council|the council's talk page]].
* Later this week, it will be possible for communities to configure [[mw:Special:MyLanguage/Help:Edit check|Edit Checks and Suggestions]] so that they are [[mw:Special:MyLanguage/Edit check/Configuration#extraNamespaces|shown within additional namespaces]], such as a &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Draft:&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; namespace. [[mw:Talk:VisualEditor/Suggestion Mode|Feedback is welcome]].
* A bug in the [[m:Special:GlobalWatchlist|Global Watchlist]] is causing the page to fail to load for users that have unseen changes on Wikimedia Commons. Developers are working on a fix. Until then, affected users can follow the workarounds described in the [[mw:Special:MyLanguage/Extension:GlobalWatchlist#GlobalWatchlist temporarily broken for users with Wikimedia Commons|relevant]] help section.
* New magic words &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{CATEGORYSORT:TIMESTAMP}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{CATEGORYSORT:RTIMESTAMP}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; are now available for use. They are able to change the default sorting of categories to be based on timestamp of categorization. [https://phabricator.wikimedia.org/T433768]
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]].  For example, the issue where [https://wikistats.wmcloud.org/ Wikistats] for Wikipedias was not reachable has now been fixed. [https://phabricator.wikimedia.org/T435959]

'''Updates for technical contributors'''
* The datacenter switchover, scheduled on September 23, has been postponed. [[diffblog:2025/03/12/hear-that-the-wikis-go-silent-twice-a-year/|The equinox exercise]] revealed capacity issues, preventing the switch of all services to the other datacenter. The process has been paused, prioritizing investigating that issue, to ensure that we continue to be able to serve our users reliably. A new exercise will be scheduled. Meanwhile, all traffic and edits continue to work as usual.
* On September 23, a power loss briefly affected all wikis, creating intermittent issues on both reading and editing modes. This also affected Gerrit. This is unrelated to the datacenter switchover postponement. [https://www.wikimediastatus.net/incidents/9fm19t8qykk8]
* [[mw:Special:MyLanguage/Help:Extension:Produnto|Produnto]] has been deployed to mediawiki.org and [[phab:T421436|some Indic language wikis]]. We want to hear your feedback on this experimental product. Produnto allows users to use Lua modules hosted in [[gitlab:repos/lua|Wikimedia's GitLab]]. It simplifies the sharing of Lua code between wikis. Instead of copying each individual Lua module from one wiki to another, users will be able to easily share packages across all wikis.
* [[File:Reload icon with two arrows.svg|12px|link=|class=skin-invert|Recurrent item]] Detailed code updates later this week: [[mw:MediaWiki 1.47/wmf.22|MediaWiki]]

'''Meetings and events'''
* A [[m:Event:Cross-wiki Code Collaboration Workshop|Cross-wiki code collaboration workshop]] will be held online on October 2 at 12:30 UTC, bringing together South Asian technical contributors to learn about Produnto and begin piloting it on Hindi, Punjabi, Odia, Telugu, and Malayalam Wikipedias. Produnto is a package manager for deploying Lua modules hosted on GitLab to Wikimedia wikis, and was recently deployed on some pilot wikis.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2026/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2026-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 10:49, 28 September 2026 (UTC)
&lt;!-- Message sent by User:STei (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=31092225 --&gt;</text>
      <sha1>spf1wqu3bjdc9im27bdk80v7sx2krh3</sha1>
    </revision>
  </page>
  <page>
    <title>Solving simultaneous equations</title>
    <ns>0</ns>
    <id>266023</id>
    <revision>
      <id>2834820</id>
      <parentid>2833755</parentid>
      <timestamp>2026-09-28T08:44:42Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Solving 2 by 3 */</comment>
      <origin>2834820</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79204" sha1="i504b22lbvccjeqqvv89htvnq7s48sm" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() :
  error detected in processing.
    (&lt;class 'ValueError'&gt;, ValueError('too many values to unpack (expected 3)',))
  input =
    [3, 2, -13, 7]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input) :
    '''
input represents system:
a1*x + b1*y + c1*z + d1 = 0 ... (1)
a2*x + b2*y + c2*z + d2 = 0 ... (2)
a3*x + b3*y + c3*z + d3 = 0 ... (3)

input = (
(a1,b1,c1,d1),
(a2,b2,c2,d2),
(a3,b3,c3,d3),
)

To invoke:
x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
'''
    def checkDirectionNumbers (input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3
# The direction numbers of the lines of intersection.
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 2 and 3.')
            return
        dn2 = [b3*c1-b1*c3, a1*c3-a3*c1, b1*a3-b3*a1]
        if dn2[0] == dn2[1] == dn2[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 3.')
            return
        dn3 = [b1*c2-b2*c1, a2*c1-a1*c2, b2*a1-b1*a2]
        if dn3[0] == dn3[1] == dn3[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 2.')
            return
# Are 2 lines of intersection parallel?
# If so, and no 2 planes are parallel, the 3 planes form a tent.
        a2,b2,c2 = dn2
        a3,b3,c3 = dn3
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 3 rows form a tent.');
            return

    def solve3by4_(input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3

        if (a1 == b1 == c1 == 0) or (a2 == b2 == c2 == 0) or (a3 == b3 == c3 == 0) :
            print ('solve3by4() : empty row in input.')
            return None
        if (a1 == a2 == a3 == 0) or (b1 == b2 == b3 == 0) or (c1 == c2 == c3 == 0) :
            print ('solve3by4(). empty column in input.')
            return None
        if (d1 == d2 == d3 == 0) :
            # This empty column means that trivial solution is valid.
            return 0,0,0

# Sort input if necessary so that a1 is non-zero.
        if a1 : pass
        elif a2 :
            row1,row2 = row2,row1
            a1,b1,c1,d1 = row1
            a2,b2,c2,d2 = row2
        else :
            row1,row3 = row3,row1
            a1,b1,c1,d1 = row1
            a3,b3,c3,d3 = row3

# The matrix is:
# a1, b1, c1, d1 with a1 non-zero.
# a2, b2, c2, d2
# a3, b3, c3, d3

# Process rows, if necessary, so that a2,a3 become zero.

        if a2 :
            L1 = [a2*v for v in row1]
            L2 = [a1*v for v in row2]
            zero,b2_,c2_,d2_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b2_,c2_,d2_ =  a2,b2,c2,d2

        if a3 :
            L1 = [a3*v for v in row1]
            L2 = [a1*v for v in row3]
            zero,b3_,c3_,d3_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b3_,c3_,d3_ =  a3,b3,c3,d3

# The matrix is:
# a1, b1,  c1,  d1 with a1 non-zero.
# 0,  b2_, c2_, d2_
# 0,  b3_, c3_, d3_
        data1 = (
(b2_, c2_, d2_),
(b3_, c3_, d3_),
)
        data2 = solve2by3(data1)
        if data2 == None :
            checkDirectionNumbers (input) # If solve2by3() fails, this line shows why.
            return None
        y,z = data2
        # a1*x + b1*y + c1*z + d1 = 0
        # a1*x = -( b1*y + c1*z + d1 )
        x = -(b1*y + c1*z + d1)/a1 # Here is why a1 must be non-zero.

        return x,y,z

    output = error = ''
    try : output = solve3by4_(input)
    except : error = sys.exc_info()[:2]

    if (output == None) or error :
        if error :
            print ('solve3by4() :')
            print ('  error detected in processing.')
            print ('   ', error)
        print ('  input of solve3by4() =')
        print ('   ', input[0])
        print ('   ', input[1])
        print ('   ', input[2])
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>i504b22lbvccjeqqvv89htvnq7s48sm</sha1>
    </revision>
    <revision>
      <id>2834821</id>
      <parentid>2834820</parentid>
      <timestamp>2026-09-28T08:55:39Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Solving 2 by 3 */</comment>
      <origin>2834821</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79241" sha1="bc0045bkxfewmsnm0b6qswbgk31klsn" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() :
  error detected in processing.
    (&lt;class 'ValueError'&gt;, ValueError('too many values to unpack (expected 3)',))
  input =
    [3, 2, -13, 7]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input) :
    '''
input represents system:
a1*x + b1*y + c1*z + d1 = 0 ... (1)
a2*x + b2*y + c2*z + d2 = 0 ... (2)
a3*x + b3*y + c3*z + d3 = 0 ... (3)

input = (
(a1,b1,c1,d1),
(a2,b2,c2,d2),
(a3,b3,c3,d3),
)

To invoke:
x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
'''
    def checkDirectionNumbers (input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3
# The direction numbers of the lines of intersection.
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 2 and 3.')
            return
        dn2 = [b3*c1-b1*c3, a1*c3-a3*c1, b1*a3-b3*a1]
        if dn2[0] == dn2[1] == dn2[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 3.')
            return
        dn3 = [b1*c2-b2*c1, a2*c1-a1*c2, b2*a1-b1*a2]
        if dn3[0] == dn3[1] == dn3[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 2.')
            return
# Are 2 lines of intersection parallel?
# If so, and no 2 planes are parallel, the 3 planes form a tent.
        a2,b2,c2 = dn2
        a3,b3,c3 = dn3
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 3 rows form a tent.');
            return

    def solve3by4_(input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3

        if (a1 == b1 == c1 == 0) or (a2 == b2 == c2 == 0) or (a3 == b3 == c3 == 0) :
            print ('solve3by4() : empty row in input.')
            return None
        if (a1 == a2 == a3 == 0) or (b1 == b2 == b3 == 0) or (c1 == c2 == c3 == 0) :
            print ('solve3by4(). empty column in input.')
            return None
        if (d1 == d2 == d3 == 0) :
            # This empty column means that trivial solution is valid.
            return 0,0,0

# Sort input if necessary so that a1 is non-zero.
        if a1 : pass
        elif a2 :
            row1,row2 = row2,row1
            a1,b1,c1,d1 = row1
            a2,b2,c2,d2 = row2
        else :
            row1,row3 = row3,row1
            a1,b1,c1,d1 = row1
            a3,b3,c3,d3 = row3

# The matrix is:
# a1, b1, c1, d1 with a1 non-zero.
# a2, b2, c2, d2
# a3, b3, c3, d3

# Process rows, if necessary, so that a2,a3 become zero.

        if a2 :
            L1 = [a2*v for v in row1]
            L2 = [a1*v for v in row2]
            zero,b2_,c2_,d2_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b2_,c2_,d2_ =  a2,b2,c2,d2

        if a3 :
            L1 = [a3*v for v in row1]
            L2 = [a1*v for v in row3]
            zero,b3_,c3_,d3_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b3_,c3_,d3_ =  a3,b3,c3,d3

# The matrix is:
# a1, b1,  c1,  d1 with a1 non-zero.
# 0,  b2_, c2_, d2_
# 0,  b3_, c3_, d3_
        data1 = (
(b2_, c2_, d2_),
(b3_, c3_, d3_),
)
        data2 = solve2by3(data1)
        if data2 == None :
            checkDirectionNumbers (input) # If solve2by3() fails, this line shows why.
            return None
        y,z = data2
        # a1*x + b1*y + c1*z + d1 = 0
        # a1*x = -( b1*y + c1*z + d1 )
        x = -(b1*y + c1*z + d1)/a1 # Here is why a1 must be non-zero.

        return x,y,z

    output = error = ''
    try : output = solve3by4_(input)
    except : error = sys.exc_info()[:2]

    if (output == None) or error :
        if error :
            print ('solve3by4() :')
            print ('  error detected in processing.')
            print ('   ', error)
        print ('  input of solve3by4() =')
        print ('   ', input[0])
        print ('   ', input[1])
        print ('   ', input[2])
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>bc0045bkxfewmsnm0b6qswbgk31klsn</sha1>
    </revision>
    <revision>
      <id>2834822</id>
      <parentid>2834821</parentid>
      <timestamp>2026-09-28T09:02:05Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Solving 2 by 3 */</comment>
      <origin>2834822</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79298" sha1="jgjqp8tuwzcp20ozs90s3v7t0n0heqz" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() :
  error detected in processing.
    (&lt;class 'ValueError'&gt;, ValueError('too many values to unpack (expected 3)',))
  input =
    [3, 2, -13, 7]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input) :
    '''
input represents system:
a1*x + b1*y + c1*z + d1 = 0 ... (1)
a2*x + b2*y + c2*z + d2 = 0 ... (2)
a3*x + b3*y + c3*z + d3 = 0 ... (3)

input = (
(a1,b1,c1,d1),
(a2,b2,c2,d2),
(a3,b3,c3,d3),
)

To invoke:
x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
'''
    def checkDirectionNumbers (input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3
# The direction numbers of the lines of intersection.
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 2 and 3.')
            return
        dn2 = [b3*c1-b1*c3, a1*c3-a3*c1, b1*a3-b3*a1]
        if dn2[0] == dn2[1] == dn2[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 3.')
            return
        dn3 = [b1*c2-b2*c1, a2*c1-a1*c2, b2*a1-b1*a2]
        if dn3[0] == dn3[1] == dn3[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 2.')
            return
# Are 2 lines of intersection parallel?
# If so, and no 2 planes are parallel, the 3 planes form a tent.
        a2,b2,c2 = dn2
        a3,b3,c3 = dn3
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 3 rows form a tent.');
            return

    def solve3by4_(input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3

        if (a1 == b1 == c1 == 0) or (a2 == b2 == c2 == 0) or (a3 == b3 == c3 == 0) :
            print ('solve3by4() : empty row in input.')
            return None
        if (a1 == a2 == a3 == 0) or (b1 == b2 == b3 == 0) or (c1 == c2 == c3 == 0) :
            print ('solve3by4(). empty column in input.')
            return None
        if (d1 == d2 == d3 == 0) :
            # This empty column means that trivial solution is valid.
            return 0,0,0

# Sort input if necessary so that a1 is non-zero.
        if a1 : pass
        elif a2 :
            row1,row2 = row2,row1
            a1,b1,c1,d1 = row1
            a2,b2,c2,d2 = row2
        else :
            row1,row3 = row3,row1
            a1,b1,c1,d1 = row1
            a3,b3,c3,d3 = row3

# The matrix is:
# a1, b1, c1, d1 with a1 non-zero.
# a2, b2, c2, d2
# a3, b3, c3, d3

# Process rows, if necessary, so that a2,a3 become zero.

        if a2 :
            L1 = [a2*v for v in row1]
            L2 = [a1*v for v in row2]
            zero,b2_,c2_,d2_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b2_,c2_,d2_ =  a2,b2,c2,d2

        if a3 :
            L1 = [a3*v for v in row1]
            L2 = [a1*v for v in row3]
            zero,b3_,c3_,d3_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b3_,c3_,d3_ =  a3,b3,c3,d3

# The matrix is:
# a1, b1,  c1,  d1 with a1 non-zero.
# 0,  b2_, c2_, d2_
# 0,  b3_, c3_, d3_
        data1 = (
(b2_, c2_, d2_),
(b3_, c3_, d3_),
)
        data2 = solve2by3(data1)
        if data2 == None :
            checkDirectionNumbers (input) # If solve2by3() fails, this line shows why.
            return None
        y,z = data2
        # a1*x + b1*y + c1*z + d1 = 0
        # a1*x = -( b1*y + c1*z + d1 )
        x = -(b1*y + c1*z + d1)/a1 # Here is why a1 must be non-zero.

        return x,y,z

    output = error = ''
    try : output = solve3by4_(input)
    except : error = sys.exc_info()[:2]

    if (output == None) or error :
        if error :
            print ('solve3by4() :')
            print ('  error detected in processing.')
            print ('   ', error)
        print ('  input of solve3by4() =')
        print ('   ', input[0])
        print ('   ', input[1])
        print ('   ', input[2])
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>jgjqp8tuwzcp20ozs90s3v7t0n0heqz</sha1>
    </revision>
    <revision>
      <id>2834823</id>
      <parentid>2834822</parentid>
      <timestamp>2026-09-28T09:07:19Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Solving 2 by 3 */</comment>
      <origin>2834823</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79205" sha1="tj905v45bfje9c2b7sqyz5f3amccqyr" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input) :
    '''
input represents system:
a1*x + b1*y + c1*z + d1 = 0 ... (1)
a2*x + b2*y + c2*z + d2 = 0 ... (2)
a3*x + b3*y + c3*z + d3 = 0 ... (3)

input = (
(a1,b1,c1,d1),
(a2,b2,c2,d2),
(a3,b3,c3,d3),
)

To invoke:
x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
'''
    def checkDirectionNumbers (input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3
# The direction numbers of the lines of intersection.
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 2 and 3.')
            return
        dn2 = [b3*c1-b1*c3, a1*c3-a3*c1, b1*a3-b3*a1]
        if dn2[0] == dn2[1] == dn2[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 3.')
            return
        dn3 = [b1*c2-b2*c1, a2*c1-a1*c2, b2*a1-b1*a2]
        if dn3[0] == dn3[1] == dn3[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 2.')
            return
# Are 2 lines of intersection parallel?
# If so, and no 2 planes are parallel, the 3 planes form a tent.
        a2,b2,c2 = dn2
        a3,b3,c3 = dn3
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 3 rows form a tent.');
            return

    def solve3by4_(input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3

        if (a1 == b1 == c1 == 0) or (a2 == b2 == c2 == 0) or (a3 == b3 == c3 == 0) :
            print ('solve3by4() : empty row in input.')
            return None
        if (a1 == a2 == a3 == 0) or (b1 == b2 == b3 == 0) or (c1 == c2 == c3 == 0) :
            print ('solve3by4(). empty column in input.')
            return None
        if (d1 == d2 == d3 == 0) :
            # This empty column means that trivial solution is valid.
            return 0,0,0

# Sort input if necessary so that a1 is non-zero.
        if a1 : pass
        elif a2 :
            row1,row2 = row2,row1
            a1,b1,c1,d1 = row1
            a2,b2,c2,d2 = row2
        else :
            row1,row3 = row3,row1
            a1,b1,c1,d1 = row1
            a3,b3,c3,d3 = row3

# The matrix is:
# a1, b1, c1, d1 with a1 non-zero.
# a2, b2, c2, d2
# a3, b3, c3, d3

# Process rows, if necessary, so that a2,a3 become zero.

        if a2 :
            L1 = [a2*v for v in row1]
            L2 = [a1*v for v in row2]
            zero,b2_,c2_,d2_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b2_,c2_,d2_ =  a2,b2,c2,d2

        if a3 :
            L1 = [a3*v for v in row1]
            L2 = [a1*v for v in row3]
            zero,b3_,c3_,d3_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b3_,c3_,d3_ =  a3,b3,c3,d3

# The matrix is:
# a1, b1,  c1,  d1 with a1 non-zero.
# 0,  b2_, c2_, d2_
# 0,  b3_, c3_, d3_
        data1 = (
(b2_, c2_, d2_),
(b3_, c3_, d3_),
)
        data2 = solve2by3(data1)
        if data2 == None :
            checkDirectionNumbers (input) # If solve2by3() fails, this line shows why.
            return None
        y,z = data2
        # a1*x + b1*y + c1*z + d1 = 0
        # a1*x = -( b1*y + c1*z + d1 )
        x = -(b1*y + c1*z + d1)/a1 # Here is why a1 must be non-zero.

        return x,y,z

    output = error = ''
    try : output = solve3by4_(input)
    except : error = sys.exc_info()[:2]

    if (output == None) or error :
        if error :
            print ('solve3by4() :')
            print ('  error detected in processing.')
            print ('   ', error)
        print ('  input of solve3by4() =')
        print ('   ', input[0])
        print ('   ', input[1])
        print ('   ', input[2])
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>tj905v45bfje9c2b7sqyz5f3amccqyr</sha1>
    </revision>
    <revision>
      <id>2834825</id>
      <parentid>2834823</parentid>
      <timestamp>2026-09-28T09:10:37Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Solving 2 by 3 */</comment>
      <origin>2834825</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79255" sha1="ne6o0pwdq3w4pvj8eewe6qgpumhtbcv" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input) :
    '''
input represents system:
a1*x + b1*y + c1*z + d1 = 0 ... (1)
a2*x + b2*y + c2*z + d2 = 0 ... (2)
a3*x + b3*y + c3*z + d3 = 0 ... (3)

input = (
(a1,b1,c1,d1),
(a2,b2,c2,d2),
(a3,b3,c3,d3),
)

To invoke:
x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
'''
    def checkDirectionNumbers (input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3
# The direction numbers of the lines of intersection.
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 2 and 3.')
            return
        dn2 = [b3*c1-b1*c3, a1*c3-a3*c1, b1*a3-b3*a1]
        if dn2[0] == dn2[1] == dn2[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 3.')
            return
        dn3 = [b1*c2-b2*c1, a2*c1-a1*c2, b2*a1-b1*a2]
        if dn3[0] == dn3[1] == dn3[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 2.')
            return
# Are 2 lines of intersection parallel?
# If so, and no 2 planes are parallel, the 3 planes form a tent.
        a2,b2,c2 = dn2
        a3,b3,c3 = dn3
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 3 rows form a tent.');
            return

    def solve3by4_(input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3

        if (a1 == b1 == c1 == 0) or (a2 == b2 == c2 == 0) or (a3 == b3 == c3 == 0) :
            print ('solve3by4() : empty row in input.')
            return None
        if (a1 == a2 == a3 == 0) or (b1 == b2 == b3 == 0) or (c1 == c2 == c3 == 0) :
            print ('solve3by4(). empty column in input.')
            return None
        if (d1 == d2 == d3 == 0) :
            # This empty column means that trivial solution is valid.
            return 0,0,0

# Sort input if necessary so that a1 is non-zero.
        if a1 : pass
        elif a2 :
            row1,row2 = row2,row1
            a1,b1,c1,d1 = row1
            a2,b2,c2,d2 = row2
        else :
            row1,row3 = row3,row1
            a1,b1,c1,d1 = row1
            a3,b3,c3,d3 = row3

# The matrix is:
# a1, b1, c1, d1 with a1 non-zero.
# a2, b2, c2, d2
# a3, b3, c3, d3

# Process rows, if necessary, so that a2,a3 become zero.

        if a2 :
            L1 = [a2*v for v in row1]
            L2 = [a1*v for v in row2]
            zero,b2_,c2_,d2_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b2_,c2_,d2_ =  a2,b2,c2,d2

        if a3 :
            L1 = [a3*v for v in row1]
            L2 = [a1*v for v in row3]
            zero,b3_,c3_,d3_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b3_,c3_,d3_ =  a3,b3,c3,d3

# The matrix is:
# a1, b1,  c1,  d1 with a1 non-zero.
# 0,  b2_, c2_, d2_
# 0,  b3_, c3_, d3_
        data1 = (
(b2_, c2_, d2_),
(b3_, c3_, d3_),
)
        data2 = solve2by3(data1)
        if data2 == None :
            checkDirectionNumbers (input) # If solve2by3() fails, this line shows why.
            return None
        y,z = data2
        # a1*x + b1*y + c1*z + d1 = 0
        # a1*x = -( b1*y + c1*z + d1 )
        x = -(b1*y + c1*z + d1)/a1 # Here is why a1 must be non-zero.

        return x,y,z

    output = error = ''
    try : output = solve3by4_(input)
    except : error = sys.exc_info()[:2]

    if (output == None) or error :
        if error :
            print ('solve3by4() :')
            print ('  error detected in processing.')
            print ('   ', error)
        print ('  input of solve3by4() =')
        print ('   ', input[0])
        print ('   ', input[1])
        print ('   ', input[2])
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>ne6o0pwdq3w4pvj8eewe6qgpumhtbcv</sha1>
    </revision>
    <revision>
      <id>2834826</id>
      <parentid>2834825</parentid>
      <timestamp>2026-09-28T09:22:00Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Solving 2 by 3 */</comment>
      <origin>2834826</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79276" sha1="5gf5vp0evw4rt2e3xgxps2kbi6q6iuz" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input) :
    '''
input represents system:
a1*x + b1*y + c1*z + d1 = 0 ... (1)
a2*x + b2*y + c2*z + d2 = 0 ... (2)
a3*x + b3*y + c3*z + d3 = 0 ... (3)

input = (
(a1,b1,c1,d1),
(a2,b2,c2,d2),
(a3,b3,c3,d3),
)

To invoke:
x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
'''
    def checkDirectionNumbers (input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3
# The direction numbers of the lines of intersection.
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 2 and 3.')
            return
        dn2 = [b3*c1-b1*c3, a1*c3-a3*c1, b1*a3-b3*a1]
        if dn2[0] == dn2[1] == dn2[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 3.')
            return
        dn3 = [b1*c2-b2*c1, a2*c1-a1*c2, b2*a1-b1*a2]
        if dn3[0] == dn3[1] == dn3[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 2.')
            return
# Are 2 lines of intersection parallel?
# If so, and no 2 planes are parallel, the 3 planes form a tent.
        a2,b2,c2 = dn2
        a3,b3,c3 = dn3
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 3 rows form a tent.');
            return

    def solve3by4_(input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3

        if (a1 == b1 == c1 == 0) or (a2 == b2 == c2 == 0) or (a3 == b3 == c3 == 0) :
            print ('solve3by4() : empty row in input.')
            return None
        if (a1 == a2 == a3 == 0) or (b1 == b2 == b3 == 0) or (c1 == c2 == c3 == 0) :
            print ('solve3by4(). empty column in input.')
            return None
        if (d1 == d2 == d3 == 0) :
            # This empty column means that trivial solution is valid.
            return 0,0,0

# Sort input if necessary so that a1 is non-zero.
        if a1 : pass
        elif a2 :
            row1,row2 = row2,row1
            a1,b1,c1,d1 = row1
            a2,b2,c2,d2 = row2
        else :
            row1,row3 = row3,row1
            a1,b1,c1,d1 = row1
            a3,b3,c3,d3 = row3

# The matrix is:
# a1, b1, c1, d1 with a1 non-zero.
# a2, b2, c2, d2
# a3, b3, c3, d3

# Process rows, if necessary, so that a2,a3 become zero.

        if a2 :
            L1 = [a2*v for v in row1]
            L2 = [a1*v for v in row2]
            zero,b2_,c2_,d2_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b2_,c2_,d2_ =  a2,b2,c2,d2

        if a3 :
            L1 = [a3*v for v in row1]
            L2 = [a1*v for v in row3]
            zero,b3_,c3_,d3_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b3_,c3_,d3_ =  a3,b3,c3,d3

# The matrix is:
# a1, b1,  c1,  d1 with a1 non-zero.
# 0,  b2_, c2_, d2_
# 0,  b3_, c3_, d3_
        data1 = (
(b2_, c2_, d2_),
(b3_, c3_, d3_),
)
        data2 = solve2by3(data1)
        if data2 == None :
            checkDirectionNumbers (input) # If solve2by3() fails, this line shows why.
            return None
        y,z = data2
        # a1*x + b1*y + c1*z + d1 = 0
        # a1*x = -( b1*y + c1*z + d1 )
        x = -(b1*y + c1*z + d1)/a1 # Here is why a1 must be non-zero.

        return x,y,z

    output = error = ''
    try : output = solve3by4_(input)
    except : error = sys.exc_info()[:2]

    if (output == None) or error :
        if error :
            print ('solve3by4() :')
            print ('  error detected in processing.')
            print ('   ', error)
        print ('  input of solve3by4() =')
        print ('   ', input[0])
        print ('   ', input[1])
        print ('   ', input[2])
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>5gf5vp0evw4rt2e3xgxps2kbi6q6iuz</sha1>
    </revision>
    <revision>
      <id>2834827</id>
      <parentid>2834826</parentid>
      <timestamp>2026-09-28T09:25:25Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Solving 3 by 4 */</comment>
      <origin>2834827</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79296" sha1="5wibskltxzvmr72vfdd4z14xs09m5yx" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Implementation===
&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input) :
    '''
input represents system:
a1*x + b1*y + c1*z + d1 = 0 ... (1)
a2*x + b2*y + c2*z + d2 = 0 ... (2)
a3*x + b3*y + c3*z + d3 = 0 ... (3)

input = (
(a1,b1,c1,d1),
(a2,b2,c2,d2),
(a3,b3,c3,d3),
)

To invoke:
x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
'''
    def checkDirectionNumbers (input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3
# The direction numbers of the lines of intersection.
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 2 and 3.')
            return
        dn2 = [b3*c1-b1*c3, a1*c3-a3*c1, b1*a3-b3*a1]
        if dn2[0] == dn2[1] == dn2[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 3.')
            return
        dn3 = [b1*c2-b2*c1, a2*c1-a1*c2, b2*a1-b1*a2]
        if dn3[0] == dn3[1] == dn3[2] == 0 :
            print ('checkDirectionNumbers() : 2 rows parallel, 1 and 2.')
            return
# Are 2 lines of intersection parallel?
# If so, and no 2 planes are parallel, the 3 planes form a tent.
        a2,b2,c2 = dn2
        a3,b3,c3 = dn3
        dn1 = [b3*c2-b2*c3, a2*c3-a3*c2, b2*a3-b3*a2]
        if dn1[0] == dn1[1] == dn1[2] == 0 :
            print ('checkDirectionNumbers() : 3 rows form a tent.');
            return

    def solve3by4_(input) :
        row1,row2,row3 = input
        a1,b1,c1,d1 = row1
        a2,b2,c2,d2 = row2
        a3,b3,c3,d3 = row3

        if (a1 == b1 == c1 == 0) or (a2 == b2 == c2 == 0) or (a3 == b3 == c3 == 0) :
            print ('solve3by4() : empty row in input.')
            return None
        if (a1 == a2 == a3 == 0) or (b1 == b2 == b3 == 0) or (c1 == c2 == c3 == 0) :
            print ('solve3by4(). empty column in input.')
            return None
        if (d1 == d2 == d3 == 0) :
            # This empty column means that trivial solution is valid.
            return 0,0,0

# Sort input if necessary so that a1 is non-zero.
        if a1 : pass
        elif a2 :
            row1,row2 = row2,row1
            a1,b1,c1,d1 = row1
            a2,b2,c2,d2 = row2
        else :
            row1,row3 = row3,row1
            a1,b1,c1,d1 = row1
            a3,b3,c3,d3 = row3

# The matrix is:
# a1, b1, c1, d1 with a1 non-zero.
# a2, b2, c2, d2
# a3, b3, c3, d3

# Process rows, if necessary, so that a2,a3 become zero.

        if a2 :
            L1 = [a2*v for v in row1]
            L2 = [a1*v for v in row2]
            zero,b2_,c2_,d2_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b2_,c2_,d2_ =  a2,b2,c2,d2

        if a3 :
            L1 = [a3*v for v in row1]
            L2 = [a1*v for v in row3]
            zero,b3_,c3_,d3_ = [ L1[p]-L2[p] for p in (0,1,2,3) ]
        else : zero,b3_,c3_,d3_ =  a3,b3,c3,d3

# The matrix is:
# a1, b1,  c1,  d1 with a1 non-zero.
# 0,  b2_, c2_, d2_
# 0,  b3_, c3_, d3_
        data1 = (
(b2_, c2_, d2_),
(b3_, c3_, d3_),
)
        data2 = solve2by3(data1)
        if data2 == None :
            checkDirectionNumbers (input) # If solve2by3() fails, this line shows why.
            return None
        y,z = data2
        # a1*x + b1*y + c1*z + d1 = 0
        # a1*x = -( b1*y + c1*z + d1 )
        x = -(b1*y + c1*z + d1)/a1 # Here is why a1 must be non-zero.

        return x,y,z

    output = error = ''
    try : output = solve3by4_(input)
    except : error = sys.exc_info()[:2]

    if (output == None) or error :
        if error :
            print ('solve3by4() :')
            print ('  error detected in processing.')
            print ('   ', error)
        print ('  input of solve3by4() =')
        print ('   ', input[0])
        print ('   ', input[1])
        print ('   ', input[2])
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>5wibskltxzvmr72vfdd4z14xs09m5yx</sha1>
    </revision>
    <revision>
      <id>2834832</id>
      <parentid>2834827</parentid>
      <timestamp>2026-09-28T10:42:15Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Implementation */</comment>
      <origin>2834832</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79728" sha1="8jgwt2o5tyzfxwesxdwkiubv0z611hl" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Implementation===
&lt;syntaxhighlight lang=python&gt;
def two_planes_parallel (plane1, plane2) :
    '''
status = two_planes_parallel (plane1, plane2)
status = True if parallel.
    '''
    A1,B1,C1 = plane1[:3]
    A2,B2,C2 = plane2[:3]
    
    A3 = B1*C2 - B2*C1
    B3 = C1*A2 - C2*A1
    C3 = A1*B2 - A2*B1

    return ({ A3,B3,C3 } == {0})


def check_three_rows (input) :
    '''
result = check_three_rows (input)
result is None or string. 
    '''
    row0,row1,row2 = input

    if two_planes_parallel (row0,row1) : return 'Two rows parallel: 0,1'
    if two_planes_parallel (row0,row2) : return 'Two rows parallel: 0,2'
    if two_planes_parallel (row1,row2) : return 'Two rows parallel: 1,2'

    A0,B0,C0 = row0[:3] ; A1,B1,C1 = row1[:3] ; A2,B2,C2 = row2[:3]
    # Line at intersection of planes 1,2 has direction numbers:
    A3 = B1*C2 - B2*C1 ; B3 = C1*A2 - C2*A1 ; C3 = A1*B2 - A2*B1
    # Line at intersection of planes 0,2 has direction numbers:
    A4 = B0*C2 - B2*C0 ; B4 = C0*A2 - C2*A0 ; C4 = A0*B2 - A2*B0
    # Are 2 lines of intersection parallel?
    if two_planes_parallel ((A3,B3,C3), (A4,B4,C4)) :
        return 'Three rows form tent.'
    return None


def solve3by4 (input_, level=0) :
    '''
input_ is :
    [ [a0,b0,c0,d0],
      [a1,b1,c1,d1],
      [a2,b2,c2,d2] ]
    
representing system:
    a0*x + b0*y + c0*z + d0 = 0
    a1*x + b1*y + c1*z + d1 = 0
    a2*x + b2*y + c2*z + d2 = 0

x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve3by4 (input_, level = {}) :'.format(level)
    else : thisName = 'solve3by4 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        twelve_values = a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2
        for v in twelve_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in twelve_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2) = [ dD(str(v)) for v in twelve_values ]
        #
        a0 or b0 or c0 or {}['Row 0 empty.'] ; a0 or a1 or a2 or {}['Column 0 empty.']
        a1 or b1 or c1 or {}['Row 1 empty.'] ; b0 or b1 or b2 or {}['Column 1 empty.']
        a2 or b2 or c2 or {}['Row 2 empty.'] ; c0 or c1 or c2 or {}['Column 2 empty.']
        #
        input = [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ]
        result = check_three_rows (input) ; result and {}[result]
        #
        if not a0 :
            # Make a0 non-zero.
            if a1 : input[0],input[1] = input[1],input[0]
            else  : input[0],input[2] = input[2],input[0]
        #
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
# a1, b1, c1, d1
# a2, b2, c2, d2
        # Process rows so that a1, a2 become zero.

        row0 = input[0] ; a0 = row0[0]
        for p in (1,2) :
            rowp = input[p] ; ap = rowp[0]
            if ap :
                if   ap == a0 : input[p] = [ (v1-v2) for (v1,v2) in zip(row0, rowp) ]
                elif ap == -a0: input[p] = [ (v1+v2) for (v1,v2) in zip(row0, rowp) ]
                else : input[p] = [ (v0*ap-vp*a0) for (v0,vp) in zip(row0, rowp) ]
                if input[p][0] : {}['Internal error. input[{}][0] should be 0.'.format(p)] 

        (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
#  0, b1, c1, d1
#  0, b2, c2, d2
        two_rows = (b1,c1,d1), (b2,c2,d2) 
        output = solve2by3 (two_rows, level-1)
        if output :
            y,z = output
            # a0 x + b0 y + c0 z + d0 = 0
            x = -(b0*y + c0*z + d0)/a0 # Here is why a0 must be non-zero.
            output = x,y,z
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve3by4() =')
            print ('   ', input[0])
            print ('   ', input[1])
            print ('   ', input[2])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>8jgwt2o5tyzfxwesxdwkiubv0z611hl</sha1>
    </revision>
    <revision>
      <id>2834836</id>
      <parentid>2834832</parentid>
      <timestamp>2026-09-28T11:22:07Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Implementation */</comment>
      <origin>2834836</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79801" sha1="5zu12h5ita8yf30sr2f6gnj1tpaph6b" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Auxiliary functions===
&lt;syntaxhighlight lang=python&gt;
def two_planes_parallel (plane1, plane2) :
    '''
status = two_planes_parallel (plane1, plane2)
status = True if parallel.
    '''
    A1,B1,C1 = plane1[:3]
    A2,B2,C2 = plane2[:3]
    
    A3 = B1*C2 - B2*C1
    B3 = C1*A2 - C2*A1
    C3 = A1*B2 - A2*B1

    return ({ A3,B3,C3 } == {0})


def check_three_rows (input) :
    '''
result = check_three_rows (input)
result is None or string. 
    '''
    row0,row1,row2 = input

    if two_planes_parallel (row0,row1) : return 'Two rows parallel: 0,1'
    if two_planes_parallel (row0,row2) : return 'Two rows parallel: 0,2'
    if two_planes_parallel (row1,row2) : return 'Two rows parallel: 1,2'

    A0,B0,C0 = row0[:3] ; A1,B1,C1 = row1[:3] ; A2,B2,C2 = row2[:3]
    # Line at intersection of planes 1,2 has direction numbers:
    A3 = B1*C2 - B2*C1 ; B3 = C1*A2 - C2*A1 ; C3 = A1*B2 - A2*B1
    # Line at intersection of planes 0,2 has direction numbers:
    A4 = B0*C2 - B2*C0 ; B4 = C0*A2 - C2*A0 ; C4 = A0*B2 - A2*B0
    # Are 2 lines of intersection parallel?
    if two_planes_parallel ((A3,B3,C3), (A4,B4,C4)) :
        return 'Three rows form tent.'
    return None
&lt;/syntaxhighlight&gt;
===Implementation===
&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input_, level=0) :
    '''
input_ is :
    [ [a0,b0,c0,d0],
      [a1,b1,c1,d1],
      [a2,b2,c2,d2] ]
    
representing system:
    a0*x + b0*y + c0*z + d0 = 0
    a1*x + b1*y + c1*z + d1 = 0
    a2*x + b2*y + c2*z + d2 = 0

x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve3by4 (input_, level = {}) :'.format(level)
    else : thisName = 'solve3by4 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        twelve_values = a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2
        for v in twelve_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in twelve_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2) = [ dD(str(v)) for v in twelve_values ]
        #
        a0 or b0 or c0 or {}['Row 0 empty.'] ; a0 or a1 or a2 or {}['Column 0 empty.']
        a1 or b1 or c1 or {}['Row 1 empty.'] ; b0 or b1 or b2 or {}['Column 1 empty.']
        a2 or b2 or c2 or {}['Row 2 empty.'] ; c0 or c1 or c2 or {}['Column 2 empty.']
        #
        input = [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ]
        result = check_three_rows (input) ; result and {}[result]
        #
        if not a0 :
            # Make a0 non-zero.
            if a1 : input[0],input[1] = input[1],input[0]
            else  : input[0],input[2] = input[2],input[0]
        #
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
# a1, b1, c1, d1
# a2, b2, c2, d2
        # Process rows so that a1, a2 become zero.

        row0 = input[0] ; a0 = row0[0]
        for p in (1,2) :
            rowp = input[p] ; ap = rowp[0]
            if ap :
                if   ap == a0 : input[p] = [ (v1-v2) for (v1,v2) in zip(row0, rowp) ]
                elif ap == -a0: input[p] = [ (v1+v2) for (v1,v2) in zip(row0, rowp) ]
                else : input[p] = [ (v0*ap-vp*a0) for (v0,vp) in zip(row0, rowp) ]
                if input[p][0] : {}['Internal error. input[{}][0] should be 0.'.format(p)] 

        (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
#  0, b1, c1, d1
#  0, b2, c2, d2
        two_rows = (b1,c1,d1), (b2,c2,d2) 
        output = solve2by3 (two_rows, level-1)
        if output :
            y,z = output
            # a0 x + b0 y + c0 z + d0 = 0
            x = -(b0*y + c0*z + d0)/a0 # Here is why a0 must be non-zero.
            output = x,y,z
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve3by4() =')
            print ('   ', input[0])
            print ('   ', input[1])
            print ('   ', input[2])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>5zu12h5ita8yf30sr2f6gnj1tpaph6b</sha1>
    </revision>
    <revision>
      <id>2834837</id>
      <parentid>2834836</parentid>
      <timestamp>2026-09-28T11:24:08Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Implementation */</comment>
      <origin>2834837</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79817" sha1="a0laue5iqajiap4j12dgkrgfb6tll6z" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Auxiliary functions===
&lt;syntaxhighlight lang=python&gt;
def two_planes_parallel (plane1, plane2) :
    '''
status = two_planes_parallel (plane1, plane2)
status = True if parallel.
    '''
    A1,B1,C1 = plane1[:3]
    A2,B2,C2 = plane2[:3]
    
    A3 = B1*C2 - B2*C1
    B3 = C1*A2 - C2*A1
    C3 = A1*B2 - A2*B1

    return ({ A3,B3,C3 } == {0})


def check_three_rows (input) :
    '''
result = check_three_rows (input)
result is None or string. 
    '''
    row0,row1,row2 = input

    if two_planes_parallel (row0,row1) : return 'Two rows parallel: 0,1'
    if two_planes_parallel (row0,row2) : return 'Two rows parallel: 0,2'
    if two_planes_parallel (row1,row2) : return 'Two rows parallel: 1,2'

    A0,B0,C0 = row0[:3] ; A1,B1,C1 = row1[:3] ; A2,B2,C2 = row2[:3]
    # Line at intersection of planes 1,2 has direction numbers:
    A3 = B1*C2 - B2*C1 ; B3 = C1*A2 - C2*A1 ; C3 = A1*B2 - A2*B1
    # Line at intersection of planes 0,2 has direction numbers:
    A4 = B0*C2 - B2*C0 ; B4 = C0*A2 - C2*A0 ; C4 = A0*B2 - A2*B0
    # Are 2 lines of intersection parallel?
    if two_planes_parallel ((A3,B3,C3), (A4,B4,C4)) :
        return 'Three rows form tent.'
    return None
&lt;/syntaxhighlight&gt;
===Implementation===
&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input_, level=0) :
    '''
input_ is :
    [ [a0,b0,c0,d0],
      [a1,b1,c1,d1],
      [a2,b2,c2,d2] ]
    
representing system:
    a0*x + b0*y + c0*z + d0 = 0
    a1*x + b1*y + c1*z + d1 = 0
    a2*x + b2*y + c2*z + d2 = 0

x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve3by4 (input_, level = {}) :'.format(level)
    else : thisName = 'solve3by4 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        twelve_values = a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2
        for v in twelve_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in twelve_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2) = [ dD(str(v)) for v in twelve_values ]
        #
        a0 or b0 or c0 or {}['Row 0 empty.'] ; a0 or a1 or a2 or {}['Column 0 empty.']
        a1 or b1 or c1 or {}['Row 1 empty.'] ; b0 or b1 or b2 or {}['Column 1 empty.']
        a2 or b2 or c2 or {}['Row 2 empty.'] ; c0 or c1 or c2 or {}['Column 2 empty.']
        #
        input = [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ]
        result = check_three_rows (input) ; result and {}[result]
        #
        if not a0 :
            # Make a0 non-zero.
            if a1 : input[0],input[1] = input[1],input[0]
            else  : input[0],input[2] = input[2],input[0]
        #
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
# a1, b1, c1, d1
# a2, b2, c2, d2
        # Process rows so that a1, a2 become zero.

        row0 = input[0] ; a0 = row0[0]
        for p in (1,2) :
            rowp = input[p] ; ap = rowp[0]
            if ap :
                if   ap == a0 : input[p] = [ (v1-v2) for (v1,v2) in zip(row0, rowp) ]
                elif ap == -a0: input[p] = [ (v1+v2) for (v1,v2) in zip(row0, rowp) ]
                else : input[p] = [ (v0*ap-vp*a0) for (v0,vp) in zip(row0, rowp) ]
                if input[p][0] : {}['Internal error. input[{}][0] should be 0.'.format(p)] 

        (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
#  0, b1, c1, d1
#  0, b2, c2, d2
        two_rows = (b1,c1,d1), (b2,c2,d2) 
        output = solve2by3 (two_rows, level-1)
        if output :
            y,z = output
            # a0 x + b0 y + c0 z + d0 = 0
            x = -(b0*y + c0*z + d0)/a0 # Here is why a0 must be non-zero.
            output = x,y,z
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve3by4() =')
            print ('   ', input[0])
            print ('   ', input[1])
            print ('   ', input[2])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
====Examples====
Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-692, 346, -829)
    (396, -198, 517)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>a0laue5iqajiap4j12dgkrgfb6tll6z</sha1>
    </revision>
    <revision>
      <id>2834838</id>
      <parentid>2834837</parentid>
      <timestamp>2026-09-28T11:27:24Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Examples */</comment>
      <origin>2834838</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79766" sha1="prddfjtrde49u36b0nry1bfsl4zvuo6" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Auxiliary functions===
&lt;syntaxhighlight lang=python&gt;
def two_planes_parallel (plane1, plane2) :
    '''
status = two_planes_parallel (plane1, plane2)
status = True if parallel.
    '''
    A1,B1,C1 = plane1[:3]
    A2,B2,C2 = plane2[:3]
    
    A3 = B1*C2 - B2*C1
    B3 = C1*A2 - C2*A1
    C3 = A1*B2 - A2*B1

    return ({ A3,B3,C3 } == {0})


def check_three_rows (input) :
    '''
result = check_three_rows (input)
result is None or string. 
    '''
    row0,row1,row2 = input

    if two_planes_parallel (row0,row1) : return 'Two rows parallel: 0,1'
    if two_planes_parallel (row0,row2) : return 'Two rows parallel: 0,2'
    if two_planes_parallel (row1,row2) : return 'Two rows parallel: 1,2'

    A0,B0,C0 = row0[:3] ; A1,B1,C1 = row1[:3] ; A2,B2,C2 = row2[:3]
    # Line at intersection of planes 1,2 has direction numbers:
    A3 = B1*C2 - B2*C1 ; B3 = C1*A2 - C2*A1 ; C3 = A1*B2 - A2*B1
    # Line at intersection of planes 0,2 has direction numbers:
    A4 = B0*C2 - B2*C0 ; B4 = C0*A2 - C2*A0 ; C4 = A0*B2 - A2*B0
    # Are 2 lines of intersection parallel?
    if two_planes_parallel ((A3,B3,C3), (A4,B4,C4)) :
        return 'Three rows form tent.'
    return None
&lt;/syntaxhighlight&gt;
===Implementation===
&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input_, level=0) :
    '''
input_ is :
    [ [a0,b0,c0,d0],
      [a1,b1,c1,d1],
      [a2,b2,c2,d2] ]
    
representing system:
    a0*x + b0*y + c0*z + d0 = 0
    a1*x + b1*y + c1*z + d1 = 0
    a2*x + b2*y + c2*z + d2 = 0

x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve3by4 (input_, level = {}) :'.format(level)
    else : thisName = 'solve3by4 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        twelve_values = a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2
        for v in twelve_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in twelve_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2) = [ dD(str(v)) for v in twelve_values ]
        #
        a0 or b0 or c0 or {}['Row 0 empty.'] ; a0 or a1 or a2 or {}['Column 0 empty.']
        a1 or b1 or c1 or {}['Row 1 empty.'] ; b0 or b1 or b2 or {}['Column 1 empty.']
        a2 or b2 or c2 or {}['Row 2 empty.'] ; c0 or c1 or c2 or {}['Column 2 empty.']
        #
        input = [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ]
        result = check_three_rows (input) ; result and {}[result]
        #
        if not a0 :
            # Make a0 non-zero.
            if a1 : input[0],input[1] = input[1],input[0]
            else  : input[0],input[2] = input[2],input[0]
        #
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
# a1, b1, c1, d1
# a2, b2, c2, d2
        # Process rows so that a1, a2 become zero.

        row0 = input[0] ; a0 = row0[0]
        for p in (1,2) :
            rowp = input[p] ; ap = rowp[0]
            if ap :
                if   ap == a0 : input[p] = [ (v1-v2) for (v1,v2) in zip(row0, rowp) ]
                elif ap == -a0: input[p] = [ (v1+v2) for (v1,v2) in zip(row0, rowp) ]
                else : input[p] = [ (v0*ap-vp*a0) for (v0,vp) in zip(row0, rowp) ]
                if input[p][0] : {}['Internal error. input[{}][0] should be 0.'.format(p)] 

        (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
#  0, b1, c1, d1
#  0, b2, c2, d2
        two_rows = (b1,c1,d1), (b2,c2,d2) 
        output = solve2by3 (two_rows, level-1)
        if output :
            y,z = output
            # a0 x + b0 y + c0 z + d0 = 0
            x = -(b0*y + c0*z + d0)/a0 # Here is why a0 must be non-zero.
            output = x,y,z
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve3by4() =')
            print ('   ', input[0])
            print ('   ', input[1])
            print ('   ', input[2])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
====Examples====
Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Three rows form tent.')
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [-34, -6, 34, -32],
    [82, 14, -82, 72],
    [-59, -45, 59, -88]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : empty column in input.
  input =
    (-16, 0, -176)
    (-1176, 0, -1104)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-34, -6, 34, -32]
    [82, 14, -82, 72]
    [-59, -45, 59, -88]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (1392, -1008, -3728)
    (-464, 336, -752)
checkDirectionNumbers() : 2 rows parallel, 2 and 3.
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4() :
  error detected in processing.
    (&lt;class 'TypeError'&gt;, TypeError(&quot;unsupported operand type(s) for -: 'int' and 'str'&quot;,))
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>prddfjtrde49u36b0nry1bfsl4zvuo6</sha1>
    </revision>
    <revision>
      <id>2834839</id>
      <parentid>2834838</parentid>
      <timestamp>2026-09-28T11:32:08Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Examples */</comment>
      <origin>2834839</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79204" sha1="piv5a5mav7q3u5g97qfepoi4r85qm4b" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Auxiliary functions===
&lt;syntaxhighlight lang=python&gt;
def two_planes_parallel (plane1, plane2) :
    '''
status = two_planes_parallel (plane1, plane2)
status = True if parallel.
    '''
    A1,B1,C1 = plane1[:3]
    A2,B2,C2 = plane2[:3]
    
    A3 = B1*C2 - B2*C1
    B3 = C1*A2 - C2*A1
    C3 = A1*B2 - A2*B1

    return ({ A3,B3,C3 } == {0})


def check_three_rows (input) :
    '''
result = check_three_rows (input)
result is None or string. 
    '''
    row0,row1,row2 = input

    if two_planes_parallel (row0,row1) : return 'Two rows parallel: 0,1'
    if two_planes_parallel (row0,row2) : return 'Two rows parallel: 0,2'
    if two_planes_parallel (row1,row2) : return 'Two rows parallel: 1,2'

    A0,B0,C0 = row0[:3] ; A1,B1,C1 = row1[:3] ; A2,B2,C2 = row2[:3]
    # Line at intersection of planes 1,2 has direction numbers:
    A3 = B1*C2 - B2*C1 ; B3 = C1*A2 - C2*A1 ; C3 = A1*B2 - A2*B1
    # Line at intersection of planes 0,2 has direction numbers:
    A4 = B0*C2 - B2*C0 ; B4 = C0*A2 - C2*A0 ; C4 = A0*B2 - A2*B0
    # Are 2 lines of intersection parallel?
    if two_planes_parallel ((A3,B3,C3), (A4,B4,C4)) :
        return 'Three rows form tent.'
    return None
&lt;/syntaxhighlight&gt;
===Implementation===
&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input_, level=0) :
    '''
input_ is :
    [ [a0,b0,c0,d0],
      [a1,b1,c1,d1],
      [a2,b2,c2,d2] ]
    
representing system:
    a0*x + b0*y + c0*z + d0 = 0
    a1*x + b1*y + c1*z + d1 = 0
    a2*x + b2*y + c2*z + d2 = 0

x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve3by4 (input_, level = {}) :'.format(level)
    else : thisName = 'solve3by4 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        twelve_values = a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2
        for v in twelve_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in twelve_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2) = [ dD(str(v)) for v in twelve_values ]
        #
        a0 or b0 or c0 or {}['Row 0 empty.'] ; a0 or a1 or a2 or {}['Column 0 empty.']
        a1 or b1 or c1 or {}['Row 1 empty.'] ; b0 or b1 or b2 or {}['Column 1 empty.']
        a2 or b2 or c2 or {}['Row 2 empty.'] ; c0 or c1 or c2 or {}['Column 2 empty.']
        #
        input = [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ]
        result = check_three_rows (input) ; result and {}[result]
        #
        if not a0 :
            # Make a0 non-zero.
            if a1 : input[0],input[1] = input[1],input[0]
            else  : input[0],input[2] = input[2],input[0]
        #
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
# a1, b1, c1, d1
# a2, b2, c2, d2
        # Process rows so that a1, a2 become zero.

        row0 = input[0] ; a0 = row0[0]
        for p in (1,2) :
            rowp = input[p] ; ap = rowp[0]
            if ap :
                if   ap == a0 : input[p] = [ (v1-v2) for (v1,v2) in zip(row0, rowp) ]
                elif ap == -a0: input[p] = [ (v1+v2) for (v1,v2) in zip(row0, rowp) ]
                else : input[p] = [ (v0*ap-vp*a0) for (v0,vp) in zip(row0, rowp) ]
                if input[p][0] : {}['Internal error. input[{}][0] should be 0.'.format(p)] 

        (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
#  0, b1, c1, d1
#  0, b2, c2, d2
        two_rows = (b1,c1,d1), (b2,c2,d2) 
        output = solve2by3 (two_rows, level-1)
        if output :
            y,z = output
            # a0 x + b0 y + c0 z + d0 = 0
            x = -(b0*y + c0*z + d0)/a0 # Here is why a0 must be non-zero.
            output = x,y,z
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve3by4() =')
            print ('   ', input[0])
            print ('   ', input[1])
            print ('   ', input[2])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
====Examples====
Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Three rows form tent.')
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Two rows parallel: 1,2')
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Non-numeric value in input.')
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>piv5a5mav7q3u5g97qfepoi4r85qm4b</sha1>
    </revision>
    <revision>
      <id>2834840</id>
      <parentid>2834839</parentid>
      <timestamp>2026-09-28T11:34:14Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Implementation */</comment>
      <origin>2834840</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79222" sha1="lelbi14kpldo3ii2u0v7j4qdpvpenq0" xml:space="preserve">==Introduction==
The word &quot;simultaneous&quot; implies that two or more equations exist in the same context at the same time.

For example:

&lt;math&gt;3x + 2y - 13 = 0\ \dots\ (1)&lt;/math&gt;

&lt;math&gt;2x - y - 4 = 0 \ \dots\ (2)&lt;/math&gt;

Equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; are 2 simultaneous equations.

When we see two simultaneous equations, it's natural to ask the question: What are the values of &lt;math&gt;x&lt;/math&gt;
and &lt;math&gt;y&lt;/math&gt; that satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;?

To calculate &lt;math&gt;(x,y)&lt;/math&gt; common to both &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt;:

&lt;math&gt;(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)&lt;/math&gt;

&lt;math&gt;(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.&lt;/math&gt;

Substitute &lt;math&gt;3&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;(2):\ 2(3) - y - 4 = 0;\ y = 2.&lt;/math&gt;

The values &lt;math&gt;(x,y)=(3,2)&lt;/math&gt; satisfy both equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt;

In cartesian coordinate geometry of 2 dimensions, point &lt;math&gt;(3,2)&lt;/math&gt; is the point of intersection
of two lines &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2).&lt;/math&gt; See Figure 1 below.

If the 2 lines are parallel, there is no point of intersection. See Figure 2 below.

The trivial solution &lt;math&gt;(0, 0)&lt;/math&gt; is a valid solution. See Figure 3 below.
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2lines.png|&lt;small&gt;Figure 1. 2 lines that intersect at point (3,2).&lt;/small&gt;
File:0726_2parallelLines.png|&lt;small&gt;Figure 2. When 2 lines are parallel, there is no point of intersection.&lt;/small&gt;
File:0729_2lines.png|&lt;small&gt;Figure 3. The trivial solution x=y=0 is a valid solution.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
The information contained in equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; may constitute a matrix:

&lt;pre&gt;
[
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/pre&gt;

In fact, in the python programming language, the following statement is valid code:

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [2, -1,  -4]
]
&lt;/syntaxhighlight&gt;

For there to be a point of intersection, &lt;code&gt;data&lt;/code&gt; must be valid.

Here are examples of invalid data:
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],  # Both lines are parallel.
  [6,  4,  -5]   # See Figure 2 above.
]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  2, -13],
  [0,  0,  -4] # Empty row, representing equation 0x + 0y - 4 = 0.
]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
data = [
  [3,  0, -13], # Empty column representing 2 parallel lines:
  [2,  0,  -4]  # 3x = 13 and 2x = 4.
]
&lt;/syntaxhighlight&gt;

The matrix above with name &quot;data&quot; is described as matrix &quot;2 by 3,&quot; meaning that it contains 2 rows with 3 columns or 3 members per row.
==Solving 2 by 3==
===Implementation===
Below is python code containing function &lt;code&gt;solve2by3 (input).&lt;/code&gt; The function accepts as input a matrix like 
&lt;code&gt;data&lt;/code&gt; above.

After the code are examples of invocation with both good data and bad data.
&lt;syntaxhighlight lang=python&gt;
import sys

import decimal
dD = decimal.Decimal

def solve2by3 (input_, level=0) :
    '''
input represents system:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0

input = (
(a1,b1,c1),
(a2,b2,c2),
)

To invoke:
x,y = solve2by3 (input)
Function solve2by3 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level)
    else : thisName = 'solve2by3 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a1,b1,c1), (a2,b2,c2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        six_values = a1,b1,c1, a2,b2,c2
        for v in six_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in six_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ]
        #
        a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.']
        a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.']
        #
        if not a1 :
            # Make a1 non-zero.
            (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1)
        #
        if a2 :
            # Make a2 zero.
            if   a2 ==  a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2
            elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2
            else :
                list1 = [ a2*v for v in (a1,b1,c1) ]
                list2 = [ a1*v for v in (a2,b2,c2) ]
                a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ]
            a2 and {}['Internal error. a2 should be 0.']
        b2 or {}['Parallel input.']
# The matrix is:
# a1, b1, c1  with a1 non-zero
#  0, b2, c2  and  b2 non-zero.
        # 0 x + b2 y + c2 = 0
        y = -c2/b2
        # a1 x + b1 y + c1 = 0
        x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero.
        output = x,y
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve2by3() =')
            print ('   ', input[0])
            print ('   ', input[1])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
====Examples====
Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],
  [2, -1,  -4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 2.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13],  # Both lines are parallel.    
  [6,  4,  -5]   # See Figure 2 above.         
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Parallel input.')
  input of solve2by3() =
    [3, 2, -13]
    [6, 4, -5]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  2, -13, 7], # Input matrix not exactly 2 by 3.
  [6,  4,  -5]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('input_ not recognized.')
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve2by3([
  [3,  0, 2],
  [6,  0, 4]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Column 2 empty.')
  input of solve2by3() =
    [3, 0, 2]
    [6, 0, 4]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the process===
====Linear Function====
[[File:1212line01.png|thumb|400px|'''
Figure 1. Diagram of line defined by 2 points.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(-5,4), (3,1).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;3x + 8y - 17 = 0.&lt;/math&gt;
]]
Examples above calculate the point of intersection of 2 lines. 

An example of reversing the process is this task:

Given two points &lt;math&gt;(x_1, y_1), (x_2, y_2)&lt;/math&gt; calculate equation of line containing these 2 points.

Equation of line is of format &lt;math&gt;Ax + By + C = 0,&lt;/math&gt;

For example: &lt;math&gt;3x + 2y + 10 = 0\ \dots\ (1).&lt;/math&gt;

Equation &lt;math&gt;(1)&lt;/math&gt; can be expressed as &lt;math&gt;6x + 4y + 20 = 0&lt;/math&gt; or &lt;math&gt;-9x - 6y - 30 = 0.&lt;/math&gt;

Also, there appear to be 3 unknowns, but only two sets of values are required to define the line.


Assume &lt;math&gt;C = 1&lt;/math&gt; and the 2 relevant equations are:

&lt;math&gt;Ax_1 + By_1 + 1 = 0&lt;/math&gt;

&lt;math&gt;Ax_2 + By_2 + 1 = 0&lt;/math&gt;


To solve for &lt;math&gt;A, B:&lt;/math&gt;

&lt;math&gt;x_1A + y_1B + 1 = 0&lt;/math&gt;

&lt;math&gt;x_2A + y_2B + 1 = 0&lt;/math&gt;

Suppose the line is defined by points &lt;math&gt;(3,1), (-5,4):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
input = [
    [3,1,1],
    [-5,4,1],
]
result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-0.17647058823529413, -0.47058823529411764)
&lt;/syntaxhighlight&gt;
Desired equation is: &lt;math&gt;-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).&lt;/math&gt;

To improve appearance of &lt;math&gt;(2):&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

print (A/B)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
0.37500000000000006
&lt;/syntaxhighlight&gt;
Let &lt;math&gt;A, B = 3, 8&lt;/math&gt; and check:
&lt;syntaxhighlight lang=python&gt;
# python code

A,B = 3,8

p1 = x1,y1 = 3,1
p2 = x2,y2 = -5,4

for v in (p1,p2) :
    x,y = v
    C = -(A*x + B*y)
    s1 = 'C'
    print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
C -17
C -17
&lt;/syntaxhighlight&gt;
Desired equation &lt;math&gt;(2)&lt;/math&gt; becomes &lt;math&gt;3x + 8y - 17 = 0\ \dots\ (3).&lt;/math&gt;

====Wrong assumptions====
[[File:1212line02.png|thumb|400px|'''
Figure 2. Diagram of line defined by 2 points and passing through origin.'''
&lt;/br&gt;
2 points are: &lt;math&gt;(3,2), (6,4).&lt;/math&gt;
&lt;/br&gt;
Equation of line is: &lt;math&gt;2x - 3y = 0.&lt;/math&gt;
&lt;/br&gt;
Non-zero value for coefficient &lt;math&gt;C&lt;/math&gt; is impossible.
]]
Sometimes you create a matrix that seems correct, but it fails to produce a result.
Here is such an example:

What is equation of line defined by points &lt;math&gt;(3, 2), (6, 4)?&lt;/math&gt;

As above assume that &lt;math&gt;C = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

p1 = x1,y1 = 3,2 
p2 = x2,y2 = 6,4
input = [
    [x1,y1,1],
    [x2,y2,1],
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
This assumption failed:
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    [3, 2, 1]
    [6, 4, 1]
None
&lt;/syntaxhighlight&gt;
Assume that &lt;math&gt;A = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

# 1(x1) + B(y1) + 1(C) = 0
# 1(x2) + B(y2) + 1(c) = 0

# y1(B) + 1(C) + x1 = 0
# y2(B) + 1(C) + x2 = 0

input = [
    [y1,1,x1], # Notice that input has been
    [y2,1,x2], # rotated left by one column.
]

result = solve2by3 (input)
print (result)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1.5, 0.0)
&lt;/syntaxhighlight&gt;
Equation of line is: &lt;math&gt;1\cdot x + (-1.5)\cdot y + 0 = 0&lt;/math&gt; or &lt;math&gt;2x - 3y = 0&lt;/math&gt;

This line includes the origin. Therefore constant &lt;math&gt;C&lt;/math&gt; cannot be non-zero.


Generally:

* Without rotation, &lt;math&gt;A, B = \text{result}&lt;/math&gt; and &lt;math&gt;C = 1.&lt;/math&gt;

* With rotation left by one column, &lt;math&gt;B, C = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt; 


In this case:
&lt;syntaxhighlight lang=python&gt;
# python code:
&gt;&gt;&gt; B,C = result ; A = 1
&gt;&gt;&gt; A,B,C
&gt;&gt;&gt; 1,-1.5,0
&lt;/syntaxhighlight&gt;


For example of big matrix that failed the first time, but was successful the second time, see
[https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network]
below.

==Solving 3 by 4==
Below is python code containing function &lt;code&gt;solve3by4 (input).&lt;/code&gt; The function accepts as input a 3 by 4 matrix like 
that described in the code comments.

In 3d coordinate geometry the equations &lt;code&gt;(1), (2), (3)&lt;/code&gt; are the equations of 3 planes and the solution of the system
is the point of intersection of the 3 planes.

Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection:

* If 2 of the planes are parallel. See Figure 1 below.

* If the 3 planes form a &quot;tent&quot; or &quot;delta.&quot; See Figure 2, 3 or 4 below. A special case of this condition occurs when size of &quot;tent&quot; is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below.
The trivial solution &lt;math&gt;(0, 0, 0)&lt;/math&gt; is valid. See Figure 6 below.
&lt;code&gt;&lt;/code&gt;
&lt;code&gt;&lt;/code&gt;
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0726_2parallelPlanes.png|&lt;small&gt;Figure 1. When 2 planes are parallel, there is no point of intersection.&lt;/small&gt;
File:0730triangularTube.png|&lt;small&gt;Figure 2. Triangular tube, 3 planes that never intersect.&lt;/small&gt;
File:0726_3planes0.png|&lt;small&gt;Figure 3. 3 planes that form a &quot;tent,&quot; view 1. There is no point of intersection.&lt;/small&gt;
File:0726_3planes1.png|&lt;small&gt;Figure 4. 3 planes that form a &quot;tent,&quot; view 2. There is no point of intersection.&lt;/small&gt;
File:1109_3planes1.png|&lt;small&gt;Figure 5. 3 planes that intersect on a common line. Points &lt;math&gt;(6,6,1),&lt;/math&gt; &lt;math&gt;(7,7,4),&lt;/math&gt; &lt;math&gt;(8,8,7)&lt;/math&gt; are on this line and are common to all 3 planes.&lt;/small&gt;
File:0729_3planes.png|&lt;small&gt;Figure 6. 3 planes that intersect at origin (0, 0, 0).&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
===Auxiliary functions===
&lt;syntaxhighlight lang=python&gt;
def two_planes_parallel (plane1, plane2) :
    '''
status = two_planes_parallel (plane1, plane2)
status = True if parallel.
    '''
    A1,B1,C1 = plane1[:3]
    A2,B2,C2 = plane2[:3]
    
    A3 = B1*C2 - B2*C1
    B3 = C1*A2 - C2*A1
    C3 = A1*B2 - A2*B1

    return ({ A3,B3,C3 } == {0})


def check_three_rows (input) :
    '''
result = check_three_rows (input)
result is None or string. 
    '''
    row0,row1,row2 = input

    if two_planes_parallel (row0,row1) : return 'Two rows parallel: 0,1'
    if two_planes_parallel (row0,row2) : return 'Two rows parallel: 0,2'
    if two_planes_parallel (row1,row2) : return 'Two rows parallel: 1,2'

    A0,B0,C0 = row0[:3] ; A1,B1,C1 = row1[:3] ; A2,B2,C2 = row2[:3]
    # Line at intersection of planes 1,2 has direction numbers:
    A3 = B1*C2 - B2*C1 ; B3 = C1*A2 - C2*A1 ; C3 = A1*B2 - A2*B1
    # Line at intersection of planes 0,2 has direction numbers:
    A4 = B0*C2 - B2*C0 ; B4 = C0*A2 - C2*A0 ; C4 = A0*B2 - A2*B0
    # Are 2 lines of intersection parallel?
    if two_planes_parallel ((A3,B3,C3), (A4,B4,C4)) :
        return 'Three rows form tent.'
    return None
&lt;/syntaxhighlight&gt;
===Implementation===
&lt;syntaxhighlight lang=python&gt;
def solve3by4 (input_, level=0) :
    '''
input_ is :
    [ [a0,b0,c0,d0],
      [a1,b1,c1,d1],
      [a2,b2,c2,d2] ]
    
representing system:
    a0*x + b0*y + c0*z + d0 = 0
    a1*x + b1*y + c1*z + d1 = 0
    a2*x + b2*y + c2*z + d2 = 0

x,y,z = solve3by4 (input)
Function solve3by4 () may return None.

input is not changed by this function.
    '''
    if level : thisName = 'solve3by4 (input_, level = {}) :'.format(level)
    else : thisName = 'solve3by4 (input_) :'
    try :
        error = 0
        try :
            input = [ list(v) for v in input_ ]
            (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
        except : input = 0
        input or {}['input_ not recognized.']
        #
        twelve_values = a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2
        for v in twelve_values :
            isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.']
        set1 = { isinstance(v, dD) for v in twelve_values }
        if len(set1) == 2 :
            # Mix of Decimal and non-Decimal in input. Make all type Decimal.
            (a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2) = [ dD(str(v)) for v in twelve_values ]
        #
        a0 or b0 or c0 or {}['Row 0 empty.'] ; a0 or a1 or a2 or {}['Column 0 empty.']
        a1 or b1 or c1 or {}['Row 1 empty.'] ; b0 or b1 or b2 or {}['Column 1 empty.']
        a2 or b2 or c2 or {}['Row 2 empty.'] ; c0 or c1 or c2 or {}['Column 2 empty.']
        #
        input = [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ]
        result = check_three_rows (input) ; result and {}[result]
        #
        if not a0 :
            # Make a0 non-zero.
            if a1 : input[0],input[1] = input[1],input[0]
            else  : input[0],input[2] = input[2],input[0]
        #
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
# a1, b1, c1, d1
# a2, b2, c2, d2
        # Process rows so that a1, a2 become zero.

        row0 = input[0] ; a0 = row0[0]
        for p in (1,2) :
            rowp = input[p] ; ap = rowp[0]
            if ap :
                if   ap == a0 : input[p] = [ (v1-v2) for (v1,v2) in zip(row0, rowp) ]
                elif ap == -a0: input[p] = [ (v1+v2) for (v1,v2) in zip(row0, rowp) ]
                else : input[p] = [ (v0*ap-vp*a0) for (v0,vp) in zip(row0, rowp) ]
                if input[p][0] : {}['Internal error. input[{}][0] should be 0.'.format(p)] 

        (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input
# The matrix is:
# a0, b0, c0, d0 with a0 non-zero.
#  0, b1, c1, d1
#  0, b2, c2, d2
        two_rows = (b1,c1,d1), (b2,c2,d2) 
        output = solve2by3 (two_rows, level-1)
        if output :
            y,z = output
            # a0 x + b0 y + c0 z + d0 = 0
            x = -(b0*y + c0*z + d0)/a0 # Here is why a0 must be non-zero.
            output = x,y,z
    except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ]
    #
    if error :
        print (thisName)
        for v in error : print (' ', v.strip())
        if input :
            print ('  input of solve3by4() =')
            print ('   ', input[0])
            print ('   ', input[1])
            print ('   ', input[2])
        return None
    #
    return output
&lt;/syntaxhighlight&gt;
====Examples====
Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [2,  -1, 3, -17],
  [1,  3, 2, -25] ,
  [3,  2, -1, -12] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(3.0, 4.0, 5.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
  [12,  -12, -5, -24],
  [-6,   15,  8,  12] ,
  [ 0,   10,  3,   0] ,
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
(2.0, 0.0, -0.0)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, -25, -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Three rows form tent.')
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, -25, -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [34, -5, -21, 30],
    [-54, -33, 63, 62],
    [18, 11, -21, 38]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Two rows parallel: 1,2')
  input of solve3by4() =
    [34, -5, -21, 30]
    [-54, -33, 63, 62]
    [18, 11, -21, 38]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solve3by4([
    [20, 16, 2, 17],
    [-77, -27, '-25', -24],
    [81, 45, 18, 43]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve3by4 (input_) :
  &lt;class 'KeyError'&gt;
  KeyError('Non-numeric value in input.')
  input of solve3by4() =
    [20, 16, 2, 17]
    [-77, -27, '-25', -24]
    [81, 45, 18, 43]
None
&lt;/pre&gt;
{{RoundBoxBottom}}

===Reversing the Process===
====Quadratic function====
[[File:1212quadratic01.png|thumb|400px|'''
Figure 1. Graph of quadratic function defined by 3 points.'''
]]
A quadratic function &lt;math&gt;y = f(x) = Ax^2 + Bx + C&lt;/math&gt; is defined by 3 points:
&lt;math&gt;(-2, 9.0), (5, 4.8), (10, 13.8).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = Ax^2 + Bx + C&lt;/math&gt;

&lt;math&gt;A, B, C&lt;/math&gt; are 3 unknowns.

&lt;math&gt;x^2(A) + x(B) + 1(C) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

x1,y1 = -2, 9
x2,y2 =  5, 4.8
x3,y3 = 10, 13.8

input = [
    [x1**2, x1, 1, -y1],
    [x2**2, x2, 1, -y2],
    [x3**2, x3, 1, -y3],
]

A, B, C = solve3by4 (input)
s1 = 'A,B,C'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
A,B,C (0.2, -1.2, 5.8)
&lt;/syntaxhighlight&gt;
&lt;math&gt;y = f(x) = 0.2x^2 - 1.2x + 5.8&lt;/math&gt;

====Wrong assumptions====
[[File:1212plane02.png|thumb|400px|'''
Figure 2. Graph of plane containing X axis.'''
&lt;/br&gt;
Because plane passes through origin, non-zero value for coefficient &lt;math&gt;D&lt;/math&gt; is impossible.
&lt;/br&gt;
Because plane is parallel to &lt;math&gt;X&lt;/math&gt; axis, non-zero value for coefficient &lt;math&gt;A&lt;/math&gt; is impossible.
]]
In 3 dimensions a plane (&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;) is defined by 3 points:
&lt;math&gt;(5, 3, -2), (-1, 12, -8), (37, 15, -10).&lt;/math&gt;

What is equation of plane?


&lt;math&gt;Ax + By + Cz + D = 0&lt;/math&gt;

&lt;math&gt;xA + yB + zC + D = 0&lt;/math&gt;

Assume that 
&lt;math&gt;D = 1.&lt;/math&gt;

&lt;math&gt;xA + yB + zC + 1 = 0&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = [
    [ 5,  3,  -2, 1],
    [-1, 12,  -8, 1],
    [37, 15, -10, 1],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : parallel input.
  input =
    (-63, 42, -6)
    (36, -24, 32)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (5, 3, -2, 1)
    (-1, 12, -8, 1)
    (37, 15, -10, 1)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
# python code.

input = [
    [ 3,  -2, 1,  5],
    [12,  -8, 1, -1],
    [15, -10, 1, 37],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
solve2by3() : empty column in input.
  input =
    (0, 9, 63)
    (0, 12, -36)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    (3, -2, 1, 5)
    (12, -8, 1, -1)
    (15, -10, 1, 37)
result None
&lt;/syntaxhighlight&gt;
Input was invalid. Rotate input left and try again.
&lt;syntaxhighlight lang=python&gt;
input = [
    [ -2, 1,  5,  3],
    [ -8, 1, -1, 12],
    [-10, 1, 37, 15],
]

result = solve3by4 (input)
s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (1.5, 0, 0)
&lt;/syntaxhighlight&gt;
Without rotation: &lt;math&gt;A, B, C = \text{result}&lt;/math&gt; and &lt;math&gt;D = 1.&lt;/math&gt;

With rotation left: &lt;math&gt;B, C, D = \text{result}&lt;/math&gt; and &lt;math&gt;A = 1.&lt;/math&gt;

With another rotation left: &lt;math&gt;C, D, A = \text{result}&lt;/math&gt; and &lt;math&gt;B = 1.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
C, D, A = (1.5, 0, 0) ; B = 1
A, B, C, D
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0, 1, 1.5, 0)
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;0x + 1y + 1.5z + 0 = 0&lt;/math&gt; or 
&lt;math&gt;2y + 3z = 0&lt;/math&gt;
=====A better method=====
{{RoundBoxTop|theme=3}}
There are four unknown values: &lt;math&gt;A,B,C,D.&lt;/math&gt;

Create matrix 4 by 5:
&lt;syntaxhighlight lang=python&gt;
# python code.

pt1 = x1,y1,z1 = (5, 3, -2)
pt2 = x2,y2,z2 = (-1, 12, -8)
pt3 = x3,y3,z3 = (37, 15, -10)

input = []

for (x_,y_,z_) in (pt1,pt2,pt3) :
    x,y,z = [ dD(v) for v in (x_,y_,z_) ]
    input += [ [ x,y,z,1,0 ] ]

# For valid plane at least one of A,B,C must be non-zero.
# For last condition let A + B + C = 5.
input += [ [ 1,1,1,0,-5 ] ]

A,B,C,D = solveMbyN(input)

print (input)
print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) )
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[[ 5,  3,  -2, 1,  0], #  5A +  3B -  2C + 1D + 0 = 0
 [-1, 12,  -8, 1,  0], # -1A + 12B -  8C + 1D + 0 = 0
 [37, 15, -10, 1,  0], # 37A + 15B - 10C + 1D + 0 = 0
 [ 1,  1,   1, 0, -5]] #  1A +  1B +  1C + 0D - 5 = 0
 
(0)x + (2)y + (3)z + (0) = 0
&lt;/syntaxhighlight&gt;
Equation of plane is: &lt;math&gt;2y + 3z = 0.&lt;/math&gt;
{{RoundBoxBottom}}

==Solving M by (M+1)==
Below is python code containing function &lt;code&gt;solveMbyN (input).&lt;/code&gt; The function accepts as input
a 4 by 5 matrix (or greater) like that described in the code comments.

This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called.

&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
Decimal = decimal.Decimal
getcontext = decimal.getcontext

def reduceRow (row) :
    '''
This function calculates the max(abs) of all values in row,
divides all values by the max(abs) and returns the result.
[27, -15, 42] becomes [27/42, -15/42, 42/42].
Very small values are converted to zero.

To invoke:
output = reduceRow (row)
output is list or None
'''
    if False in [ isinstance(v,Decimal) for v in row ] :
        print ('reduceRow: non Decimal in input.', { type(v) for v in row })
        return None
    max = sorted([ abs(v) for v in row ])[-1]
    row = [ v/max for v in row ]
    almostZero = Decimal('1e-' + str( getcontext().prec ))
    row = [ (v, Decimal(0))[abs(v) &lt; almostZero] for v in row ]
    return row

enable_reduceRow = 0

def solveMbyN (input) :
    '''
input represents system:
a0*w + b0*x + c0*y + d0*z + e0 = 0
a1*w + b1*x + c1*y + d1*z + e1 = 0
a2*w + b2*x + c2*y + d2*z + e2 = 0
a3*w + b3*x + c3*y + d3*z + e3 = 0
or greater.
This solves a matrix of M rows by N columns where N = M+1 and M &gt;= 2

input = [
(a0,b0,c0,d0,e0),
(a1,b1,c1,d1,e1),
(a2,b2,c2,d2,e2),
(a3,b3,c3,d3,e3),
]

To invoke:
w,x,y,z = solveMbyN (input)
Function solveMbyN () may return None.

If input is size 15 by 16:
v1,v2,v3, .... ,v14,v15 = solveMbyN (input)

input is not changed by this function.
'''
    def solveMbyN_(input) :
        input = [ list(v) for v in input ]
        # input is now local, hard copy of original input.
        # This ensures that original input is not changed by this function.

        numberOfRows = len(input)
        numberOfColumns = numberOfRows+1

        if numberOfRows &lt; 2 :
            print ('solveMbyN()! len(input) &lt; 2.')
            return None
        if numberOfRows == 2 : return solve2by3(input)
        if numberOfRows == 3 : return solve3by4(input)

        for row in range (0, numberOfRows) :
            if len(input[row]) != numberOfColumns :
                print ('solveMbyN()! row with bad length.')
                print ('    row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.')
                return None
            if True not in [ bool(v) for v in input[row][:-1] ] :
                print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.')
                return None

        for column in range (0, numberOfRows) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.')
                return None

        for column in (numberOfRows,) :
            sum = 0
            for row in range (0, numberOfRows) :
                sum = bool(input[row][column])
                if sum : break
            if not sum :
                # Empty column. Trivial solution is valid.
                return (0,) * numberOfRows

        if enable_reduceRow :
            # For matrices greater than 10 by 11 enable this piece of code.
            for row in range (0, numberOfRows) :
                data = reduceRow (input[row])
                if isinstance(data,list) :
                    input[row] = data

# Arrange input if necessary so that a0 is non-zero.
        if not input[0][0] :
            for p in range (1, numberOfRows) :
                if input[p][0] :
                    input[p], input[0] = input[0], input[p]
                    break
# The matrix is:
# [a0,b0,c0,d0,e0], with a0 non-zero,
# [a1,b1,c1,d1,e1],
# [a2,b2,c2,d2,e2],
# [a3,b3,c3,d3,e3],

# Process rows so that a1, a2, a3, .... become zero.
        a0 = input[0][0]
        for p in range (1, numberOfRows) :
            ap = input[p][0]
            if ap :
                L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ]
                Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ]
                L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ]
                if L_[0] :
                    # This should not happen.
                    print ('solveMbyN()! internal error 1.')
                    return None
                input[p] = L_

# The matrix is:
# [a0, b0, c0, d0, e0], with a0 non-zero,
# [ 0,b1_,c1_,d1_,e1_],
# [ 0,b2_,c2_,d2_,e2_],
# [ 0,b3_,c3_,d3_,e3_],
        data1 = [ input[p][1:] for p in range (1, numberOfRows) ]
# data1 = [
# [b1_,c1_,d1_,e1_],
# [b2_,c2_,d2_,e2_],
# [b3_,c3_,d3_,e3_] ]

        data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself.
        if data2 == None : return None
        topRow = input[0][1:]
# b0, c0, d0, e0 topRow
#  x,  y,  z     data2
        listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:]
# listOfProducts = [ b0*x, c0*y, d0*z, e0 ]
        sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1]
# sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1]
# sumOfProducts =  b0*x + c0*y + d0*z + e0
#           a0*w + b0*x + c0*y + d0*z + e0 = 0
#           a0*w + sumOfProducts = 0

        w = -sumOfProducts/a0 # Here is why a0 must be non-zero.

        return (w,) + data2

    output = error = ''
    try : output = solveMbyN_(input)
    except : error = sys.exc_info()[:2]
    if (output == None) or error :
        if error :
            print ('solveMbyN()!')
            print ('  error detected in processing.')
            print ('   ', error)
        if len(input) &gt;= 4 :
            print ('  input of solveMbyN() with',len(input),'rows =')
            for row in input : print ('   ', row)
        return None
    return output
&lt;/syntaxhighlight&gt;

Below are examples of invocation with both good data and bad data.

Invocation with good data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
L1 = [
[Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], 
[Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], 
[Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], 
[Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], 
[Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')]
]
output = solveMbyN(L1)
print (output)
&lt;/syntaxhighlight&gt;
The following results were produced with decimal precision set at 50:
&lt;pre&gt;
(
Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362')
)
&lt;/pre&gt;
{{RoundBoxBottom}}
Invocation with bad or invalid data:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
    [-4, -6, 3, -4, -8, -8],
    [1, 0, 2, 5, 1, 7],
    [9, -3, -8, 4, 7, -5],
    [1, -9, 5, -2, 5, -2],
    [1, -3, -6, -3, -6, 6]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
# With each step towards solve2by3() the length of each number in digits can theoretically double.
solve2by3() : parallel input.
  input =
    (-181440, 181440, -101088)
    (90720, -90720, 406944)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756, -1176, 0, -1872]
    [-324, -744, 240, -936]
    [-324, -384, -120, -264]
  input of solveMbyN() with 4 rows =
    [-6, 11, 16, -4, 20]
    [-66, -5, -20, -44, -92]
    [-42, 23, -12, 12, -16]
    [-18, -21, -16, -32, 16]
  input of solveMbyN() with 5 rows =
    [-4, -6, 3, -4, -8, -8]
    [1, 0, 2, 5, 1, 7]
    [9, -3, -8, 4, 7, -5]
    [1, -9, 5, -2, 5, -2]
    [1, -3, -6, -3, -6, 6]
None
&lt;/pre&gt;
Provided that numbers are small, integers, floats and Decimal objects produce the same results:
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[-4.0, -6.0, 3.0, -4.0, -8.0, -8.0],
[1.0, 0.0, 2.0, 5.0, 1.0, 7.0],
[9.0, -3.0, -8.0, 4.0, 7.0, -5.0],
[1.0, -9.0, 5.0, -2.0, 5.0, -2.0],
[1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (-181440.0, 181440.0, -101088.0)
    (90720.0, -90720.0, 406944.0)
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [-756.0, -1176.0, 0.0, -1872.0]
    [-324.0, -744.0, 240.0, -936.0]
    [-324.0, -384.0, -120.0, -264.0]
  input of solveMbyN() with 4 rows =
    [-6.0, 11.0, 16.0, -4.0, 20.0]
    [-66.0, -5.0, -20.0, -44.0, -92.0]
    [-42.0, 23.0, -12.0, 12.0, -16.0]
    [-18.0, -21.0, -16.0, -32.0, 16.0]
  input of solveMbyN() with 5 rows =
    [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0]
    [1.0, 0.0, 2.0, 5.0, 1.0, 7.0]
    [9.0, -3.0, -8.0, 4.0, 7.0, -5.0]
    [1.0, -9.0, 5.0, -2.0, 5.0, -2.0]
    [1.0, -3.0, -6.0, -3.0, -6.0, 6.0]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
output = solveMbyN([
[Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')],
[Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')],
[Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')],
[Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')],
[Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
])
print (output)
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
solve2by3() : parallel input.
  input =
    (Decimal('-181440'), Decimal('181440'), Decimal('-101088'))
    (Decimal('90720'), Decimal('-90720'), Decimal('406944'))
checkDirectionNumbers() : 3 rows form a tent.
  input of solve3by4() =
    [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')]
    [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')]
    [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')]
  input of solveMbyN() with 4 rows =
    [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')]
    [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')]
    [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')]
    [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')]
  input of solveMbyN() with 5 rows =
    [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')]
    [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')]
    [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')]
    [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')]
    [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')]
None
&lt;/pre&gt;
{{RoundBoxBottom}}
{{RoundBoxBottom}}

===Sphere===
In cartesian coordinate geometry of three dimensions a sphere is represented by equation:

&lt;math&gt;x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.&lt;/math&gt;

On the surface of a certain sphere there are 4 known points:

&lt;syntaxhighlight lang=python&gt;
# python code

point1 = (13,7,20)
point2 = (13,7,4)
point3 = (13,-17,4)
point4 = (16,4,4)
&lt;/syntaxhighlight&gt;

What is equation of sphere?

Rearrange equation of sphere to prepare for creation of input matrix:

&lt;math&gt;(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.&lt;/math&gt;

Create input matrix of size 4 by 5:

&lt;syntaxhighlight lang=python&gt;
# python code

input = []
for (x,y,z) in (point1, point2, point3, point4) :
    input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ]

print (input)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
[ (13,   7, 20, 1, 618),
  (13,   7,  4, 1, 234),
  (13, -17,  4, 1, 474),
  (16,   4,  4, 1, 288), ] # Matrix containing 4 rows with 5 members per row.
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

result = solveMbyN(input)
print (result)
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
(-8.0, 10.0, -24.0, -104.0)
&lt;/syntaxhighlight&gt;

Equation of sphere is :

&lt;math&gt;x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0&lt;/math&gt;

===Quintic function===
[[File:1212quintic01.png|thumb|400px|'''
Figure 4. Graph of quintic function defined by 6 points.'''
]]
A quintic function &lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt; is defined by 6 points:
&lt;math&gt;(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).&lt;/math&gt;


What is &lt;math&gt;f(x)?&lt;/math&gt;


&lt;math&gt;y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F&lt;/math&gt;

&lt;math&gt;A, B, C, D, E, F&lt;/math&gt; are 6 unknowns.

&lt;math&gt;x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:

pt1 = x1,y1 = -8,0
pt2 = x2,y2 = -6,0
pt3 = x3,y3 = -3,0
pt4 = x4,y4 = 1,0
pt5 = x4,y5 = 3,0
pt6 = x6,y6 = 5,10

input = []

for pt in (pt1,pt2,pt3,pt4,pt5,pt6) :
    x,y = pt
    input += [[ x**5, x**4, x**3, x**2, x, 1, -y  ]]

print (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[[-32768, 4096, -512, 64, -8, 1,   0],
 [ -7776, 1296, -216, 36, -6, 1,   0],
 [  -243,   81,  -27,  9, -3, 1,   0],
 [     1,    1,    1,  1,  1, 1,   0],
 [   243,   81,   27,  9,  3, 1,   0],
 [  3125,  625,  125, 25,  5, 1, -10]]
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result = solveMbyN (input)

s1 = 'result'
print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, 
       -0.18028846153846156,  -0.3343531468531468,   0.47202797202797203)
&lt;/syntaxhighlight&gt;
From these results the following ratios may be deduced:

&lt;math&gt;\frac{1}{A} = 915.2;\ &lt;/math&gt;
&lt;math&gt;\frac{B}{A} = 13;\ &lt;/math&gt;
&lt;math&gt;\frac{C}{A} = 25;\ &lt;/math&gt;
&lt;math&gt;\frac{D}{A} = -165;\ &lt;/math&gt;
&lt;math&gt;\frac{E}{A} = -306;\ &lt;/math&gt;
&lt;math&gt;\frac{F}{A} = 432.&lt;/math&gt;


&lt;math&gt;y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}&lt;/math&gt;

===Resistive network===
{{RoundBoxTop|theme=2}}
[[File:1127network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 8 resistors.'''
]]
A resistive network containing 8 resistors is connected as shown in Figure 1.

What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_3?&lt;/math&gt;
{{RoundBoxBottom}}
====Solve for 8 Branch Currents====
{{RoundBoxTop|theme=2}}
[[File:1127network02.png|thumb|400px|'''
Figure 2. Diagram of resistive network showing all branch currents.'''
&lt;/br&gt;
&lt;/br&gt;
At any point to which 3 or more conductors are connected, sum of currents = 0.
&lt;/br&gt;
At point &lt;math&gt;P_4&lt;/math&gt; currents &lt;math&gt;i_1, i_7&lt;/math&gt; flow towards point; current &lt;math&gt;i_4&lt;/math&gt;
flows away from point.
&lt;/br&gt;
&lt;math&gt;i_1 + i_7 - i_4 = 0.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
In any closed loop sum of voltages = 0.
&lt;/br&gt;
In loop containing points &lt;math&gt;P_1, P_2&lt;/math&gt; voltages &lt;math&gt;e_2, e_8&lt;/math&gt; are clockwise; voltage &lt;math&gt;e_3&lt;/math&gt;
is counter-clockwise.
&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e_2 + e_8 - e_3 = 0.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Direction of flow through 
&lt;math&gt;r_7, r_8&lt;/math&gt; is assumed and is subject to correction.
]]
Voltage across &lt;math&gt;r_1:&lt;/math&gt;

Using &lt;math&gt;E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.&lt;/math&gt;


Similarly:

&lt;math&gt;e_2 = r_2\cdot b.&lt;/math&gt;

&lt;math&gt;e_3 = r_3\cdot c.&lt;/math&gt;

&lt;math&gt;\dots\dots&lt;/math&gt;

&lt;math&gt;e_8 = r_8\cdot h.&lt;/math&gt;


Build a matrix size 8 by 9 which will be input to function solveMbyN().

&lt;syntaxhighlight lang=python&gt;
# Python code:

r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80

input = []

# e1 = e2 + e7
# 10a = 20b + 70g
#          10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0
input += [[r1, - r2,   0,   0,   0,   0, - r7,   0,   0]]

# e3 = e2 + e8
# 30c = 20b + 80h
#          0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0
input += [[0, - r2,   r3,   0,   0,   0,   0, - r8,   0]]

# e5 = e7 + e4
# 50e = 70g + 40d
#          0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0
input += [[0,   0,   0, - r4,   r5,   0, - r7,   0,   0]]

# e5 = e8 + e6
# 50e = 80h + 60f
#          0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0
input += [[0,   0,   0,   0,   r5, - r6,   0, - r8,   0]]

# i7 + i5 + i8 = i2
# g + e + h = b
#          0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0
input += [[0 ,- 1,   0,   0,   1,   0,   1,   1,   0]]

# i4 = i1 + i7
# d = a + g
#          -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0
input += [[-1,   0,   0,   1,   0,   0, - 1,   0,   0]]

# i6 = i3 + i8
# f = c + h
#          0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0
input += [[0,   0, - 1,   0,   0,   1,   0, - 1,   0]]

# Total input current = 1
# a + b + c = 1
#          1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0
input += [[1,   1,   1,   0,   0,   0,   0,   0, - 1]]
&lt;/syntaxhighlight&gt;
This last entry can contain almost any arbitrary values. For example:
&lt;syntaxhighlight lang=python&gt;
# Python code:

# i7 = 2.3
input += [[0,   0,   0,   0,   0,   0,   1,   0, - 2.3]]

# e5 = 1.9
input += [[0,   0,   0,   0,   r5,   0,   0,   0, - 1.9]]
&lt;/syntaxhighlight&gt;
Changing this last entry changes all values of &lt;math&gt;e_x, i_x.&lt;/math&gt; However, the ratio &lt;math&gt;\frac{e_x}{i_x} = r_x&lt;/math&gt;
remains constant.
{{RoundBoxTop|theme=3}}
input is matrix of size 8 by 9:
&lt;syntaxhighlight&gt;
[
    [10, -20,  0,   0,  0,   0, -70,   0,  0],  # e1 = e2 + e7,      or r1(i1) - r2(i2) - r7(i7) = 0
    [ 0, -20, 30,   0,  0,   0,   0, -80,  0],  # e3 = e2 + e8,      or r3(i3) - r2(i2) - r8(i8) = 0
    [ 0,   0,  0, -40, 50,   0, -70,   0,  0],  # e5 = e4 + e7,      or r5(i5) - r4(i4) - r7(i7) = 0
    [ 0,   0,  0,   0, 50, -60,   0, -80,  0],  # e5 = e6 + e8,      or r5(i5) - r6(i6) - r8(i8) = 0
    [ 0,  -1,  0,   0,  1,   0,   1,   1,  0],  # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0
    [-1,   0,  0,   1,  0,   0,  -1,   0,  0],  # i4 = i1 + i7,      or 1(i4) - 1(i1) - 1(i7) = 0
    [ 0,   0, -1,   0,  0,   1,   0,  -1,  0],  # i6 = i3 + i8,      or 1(i6) - 1(i3) - 1(i8) = 0
    [ 1,   1,  1,   0,  0,   0,   0,   0, -1]   # i1 + i2 + i3 = 1,  or 1(i1) + 1(i2) + 1(i3) - 1 = 0
]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input)

s1 = 'values_of_i' ; print(s1,eval(s1))
s1 = 'i1 + i2 + i3' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_i (0.45727389222364606,  0.3065353746543468,  0.23619073312200717, 
             0.4350171983543536,   0.31685438726647336, 0.24812841437917316, 
            -0.022256693869292507, 0.011937681257165982)
i1 + i2 + i3 1.0
&lt;/syntaxhighlight&gt;
&lt;math&gt;i_7&lt;/math&gt; has negative value. Reverse &lt;math&gt;i_7&lt;/math&gt; to make all branch currents positive.
{{RoundBoxBottom}}

====Make all Branch Currents positive====
{{RoundBoxTop|theme=2}}
[[File:1127network03.png|thumb|400px|'''
Figure 3. Diagram of resistive network showing all branch currents positive.'''
]]
Make &lt;math&gt;i_7&lt;/math&gt; positive and check results.

&lt;syntaxhighlight lang=python&gt;
# Python code

i7 *= -1

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ 
    eval(s1) 
    for s in '12345678' 
    for s1 in [ 'i' + s + '*r' + s ]
]
s1 = 'values_of_e'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e [ 4.572738922236461,  6.130707493086937,
              7.085721993660215, 17.400687934174144,
             15.842719363323669, 14.88770486275039, 
              1.5579685708504756, 0.9550145005732785]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

t1 = (
'e1+e7-e2',              # e1 + e7 should equal e2
'e2+e8-e3',              # e2 + e8 should equal e3
'e7+e5-e4',              # e7 + e5 should equal e4
'e8+e6-e5',              # e8 + e6 should equal e5

'e1+e4-(e2+e5)',         # e1 + e4 should equal e2 + e5
'e1+e4-(e3+e6)',         # e1 + e4 should equal e3 + e6

'e1+e7+e8-e3',           # e1 + e7 + e8 should equal e3
'e7+e8+e6-e4',           # e7 + e8 + e6 should equal e4

'i1-i7-i4',              # i1 should equal i7 + i4
'i6-i3-i8',              # i6 should equal i3 + i8
'i2+i7-(i8+i5)',         # i2 + i7 should equal i8 + i5
'i1+i2+i3 - (i4+i5+i6)', # Total input current should
                         # equal total output current.
)

for s1 in t1 :
    print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
e1+e7-e2             -8.881784197001252e-16
e2+e8-e3              0.0
e7+e5-e4              0.0
e8+e6-e5              0.0

e1+e4-(e2+e5)         0.0
e1+e4-(e3+e6)         0.0

e1+e7+e8-e3          -8.881784197001252e-16
e7+e8+e6-e4           0.0

i1-i7-i4             -5.551115123125783e-17
i6-i3-i8              6.938893903907228e-18
i2+i7-(i8+i5)         0.0
i1+i2+i3 - (i4+i5+i6) 0.0
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# Python code

E1 = e3+e6
I1 = i1 + i2 + i3
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e1 + e7 + e5
I2 = i4 + i5 + i6
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 21.973426856410605
R2 21.973426856410605
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2&lt;/math&gt;
{{RoundBoxBottom}}

==Solving big matrices==

A big matrix is one of size greater than 10 by 11.

With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double.
Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, 
and each value in the matrix is an int of 1 decimal digit,
function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits.

Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level
of accuracy in the result.

Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size 
of each member manageable.

Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay
careful attention to precision and accuracy of the results.

On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 
100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 
decimal digits with the decimal point in a random position, and including the time to test output over 100 rows 
to verify that all calculated values of the solution satisfy each row.

In function solveMbyN() above find line:
&lt;syntaxhighlight lang=python&gt;
if 0 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Change it to:
&lt;syntaxhighlight lang=python&gt;
if 1 : # For matrices greater than 10 by 11 enable this piece of code.
&lt;/syntaxhighlight&gt;
Then:
&lt;syntaxhighlight lang=python&gt;
from random import getrandbits
from decimal import *
import sys
getcontext().prec=50 # Set precision to 50.

numRows=100

def randomNumber():
# 1 invocation in 256 returns 0.
    if not (getrandbits(10) &amp; 0xFF) : return Decimal(0)
    return [ w
             for q in [ getrandbits(10) &amp; 1]
             for sign in [ (-1,1)[q] ]
             for numberOfDecimalPlaces in [ getrandbits(10) &amp; 0xF ]
             for s1 in [ str(getrandbits(50))[-12:] ]
             for s2 in [ '0'*12+s1 ]
             for posn in [ len(s2)-numberOfDecimalPlaces ]
             for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ]
           ][0]

data = [[ randomNumber() for p in range(numRows+1) ]
        for p in range (numRows)
       ]

datan = sorted([ v  for row in data for v in row if v &lt; 0 ])
datap = sorted([ v  for row in data for v in row if v &gt;= 0 ])
print ('A small sample of the random numbers in data:')
print ('+ve values:',len(datap), ',', datap[:2],'\n   ',datap[-2:])
print ('-ve values:',len(datan), ',', datan[:2],'\n   ',datan[-2:])

output = solveMbyN(data)

print ('A small sample of the solutions:')
print ('output[0]  =', output[0])
print ('output[19] =', output[19])
print ('output[59] =', output[59])
print ('output[99] =', output[99])
&lt;/syntaxhighlight&gt;
&lt;pre&gt;
A small sample of the random numbers in data:
+ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')]
    [Decimal('998523464710'), Decimal('999768605679')]
-ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')]
    [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')]
    
A small sample of the solutions:
output[0]  = 0.43689400018920695253771832432651757885163362928745
output[19] = 0.19867480862268284151047885891372314311902973127033
output[59] = 0.46169681824826309421940885550155275946668017604646
output[99] = -0.020598220802233733529683040085023443421113627790464
&lt;/pre&gt;
===Testing results===
Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers.
&lt;pre&gt;
[
[v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100],
.......
[v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],]
]
&lt;/pre&gt;
Data received from function solveMbyN() is an array of 100 numbers.
&lt;pre&gt;
[ s0,s1,s2, ...... s98,s99 ]
&lt;/pre&gt;
The first row is tested with the solutions by;
&lt;pre&gt;
L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ]
&lt;/pre&gt;
sum is the sum of all items in L2.

sum is appended to L1 and the process is repeated for all 100 rows.

&lt;syntaxhighlight lang=python&gt;
L1 = []
for row in data :
# a, b, c, d, e, f, g ....... # row
# u, v, w, x, y, z            # output
        L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ]
# L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g]
        sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1]
# sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y,
#        a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1]
# sum = a*u + b*v + c*w + d*x + e*y + f*z + g
# sum is the sum of all items in L2
        L1 += [abs(sum)]

L1a = sorted(L1)
print (len(L1a), L1a[0], L1a[-1])
&lt;/syntaxhighlight&gt;
Ideally sum should be zero.

In practice it's extremely unlikely that sum will be zero. My results typically gave values close to 
&lt;pre&gt;
100 4.1065149609E-38 9.6425614873452398368906E-25
&lt;/pre&gt;
The worst of the results above was 23 zeroes after the decimal point.

Better results can be achieved by increasing precision.
===Cubic Resistive Network===
{{RoundBoxTop|theme=2}}
[[File:1129network01.png|thumb|400px|'''
Figure 1. Diagram of resistive network containing 12 resistors.'''
&lt;/br&gt;
Blue arrows show assumed direction of current.
]]
A resistive network containing 12 resistors is connected as shown in Figure 1.


The network is called &quot;cubic&quot; because, in 3 dimensions, each resistor is located on one
edge of a cube.


What is resistance &lt;math&gt;R&lt;/math&gt; between points &lt;math&gt;P_1, P_2?&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# Python code:

values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}
====Solve for 12 Branch Currents====
=====Create matrix 12 by 13=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
[10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
[ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
[ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
[10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
[ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
[ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
[-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
[ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
[-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
[ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12
[ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]  # i3 = 1.6
]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
s1 = 'values_of_i'
print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
This matrix failed:
&lt;syntaxhighlight&gt;
solve2by3() : empty row in input.
  input =
    (0, 0, 0)
    (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61)
checkDirectionNumbers() : 2 rows parallel, 1 and 2.
  input of solve3by4() =
    [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0]
    [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0]
    [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32]
&lt;/syntaxhighlight&gt;

=====Try again=====
&lt;syntaxhighlight lang=python&gt;
# Python code:
input = [
  [10, 20, -30, -40,   0,  0,   0,   0,  0,    0,   0,    0,    0],  # e1 + e2 = e3 + e4
  [ 0,  0,   0,   0,  50, 60, -70, -80,  0,    0,   0,    0,    0],  # e5 + e6 = e7 + e8
  [ 0,  0,   0,  40,   0,  0,   0, -80,  0,    0, 110, -120,    0],  # e11 + e4 = e8 + e12
  [10,  0,   0,   0, -50,  0,   0,   0, 90, -100,   0,    0,    0],  # e9 + e1 = e5 + e10
  [ 0,  0,   0,   0,   1, -1,   0,   0,  0,   -1,   0,    0,    0],  # i5 = i10 + i6
  [ 0,  0,   0,   0,   0,  0,   1,  -1,  0,    0,  -1,    0,    0],  # i7 = i11 + i8
  [-1,  0,  -1,   0,   0,  0,   0,   0,  1,    0,   0,    0,    0],  # i9 = i1 + i3
  [ 0,  0,  -1,   1,   0,  0,   0,   0,  0,    0,  -1,    0,    0],  # i4 = i3 + i11
  [-1,  1,   0,   0,   0,  0,   0,   0,  0,   -1,   0,    0,    0],  # i2 = i1 + i10
  [ 0,  0,   0,   0,   0, -1,   0,  -1,  0,    0,   0,    1,    0],  # i12 = i6 + i8
 #[ 0, -1,   0,  -1,   1,  0,   1,   0,  1,    0,   0,   -1,    0],  # i9 + i5 + i7 = i2 + i4 + i12 DELETED.
  [10, 20,   0,   0,   0,  0, -70, -80, 90,    0,   0, -120,    0],  # e9 + e1 + e2 = e7 + e8 + e12 ADDED.
  [ 0,  0,   1,   0,   0,  0,   0,   0,  0,    0,   0,    0, -1.6]   # i3 = 1.6
]

import decimal
dD = decimal.Decimal
dgt = decimal.getcontext()
Precision = dgt.prec = 30
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))

# Convert all values in input to decimal objects with precision of 30.

input = [ [ dD(str(p)) for p in v  ] for v in input ]

i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
# values_of_i

5.5027171492204899777282850777     9.3809275973899113038721525414     1.5999999999999999999999999999  # i3 should be 1.6.
4.8661430860000781463681475402     6.12901340210213730316883522816    2.25080295393271597702496776458
4.68526862814050716992927753670    1.41912554214042902356112999643    7.10271714922048997772828507766
3.87821044816942132614386746358    3.26614308600007814636814754027    3.66992849607314500058609776103
&lt;/syntaxhighlight&gt;

====Compute R====
{{RoundBoxTop|theme=8}}
&lt;syntaxhighlight lang=python&gt;
# python code

values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ]
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
values_of_e
55.0271714922048997772828507770    187.618551947798226077443050828    47.9999999999999999999999999970 
194.645723440003125854725901608    306.450670105106865158441761408    135.048177235962958621498065875
327.968803969835501895049427569    113.530043371234321884890399714    639.244543429844097995545656989
387.821044816942132614386746358    359.275739460008596100496229430    440.391419528777400070331731324
&lt;/syntaxhighlight&gt;
&lt;math&gt;e_3&lt;/math&gt; should be &lt;math&gt;48,\ i_3*r_3&lt;/math&gt; or &lt;math&gt;1.6*30.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
E1 = e5 + e10 + e2
I1 = i9 + i5 + i7
R1 = E1/I1
s1 = 'R1' ; print(s1,eval(s1))

E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4
I2 = i2 + i4 + i12
R2 = E2/I2
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.2208688540148837936367036806
R2 49.2208688540148837936367036803
&lt;/syntaxhighlight&gt;
Display results with precision of 28:
&lt;syntaxhighlight lang=python&gt;
dgt.prec = 28
R1 += 0 ; R2 += 0
dgt.prec = Precision

s1 = 'R1' ; print(s1,eval(s1))
s1 = 'R2' ; print(s1,eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R1 49.22086885401488379363670368 
R2 49.22086885401488379363670368
&lt;/syntaxhighlight&gt;
&lt;math&gt;R = R_1 = R_2.&lt;/math&gt;
{{RoundBoxBottom}}

====Review====
In matrix all values &lt;math&gt;i_1, i_2, \dots, i_{11}, i_{12}&lt;/math&gt; and &lt;math&gt;e_1, e_2, \dots, e_{11}, e_{12}&lt;/math&gt; are included.


Testing was done with the following lines substituted for last line of matrix:
&lt;syntaxhighlight lang=python&gt;
# python code
[ 0,  0,   0,    0,   0,  0,   1,    0,  0,    0,   0,    0, -3.15]  # i7 = 3.15
[ 0,  0,   0,   r4,   0,  0,   0,    0,  0,    0,   0,    0, -6   ]  # e4 = 6
[ 0,  0,   r3,   0,   0,  0,   0,   r8,  0,    0,   0,    0, -5   ]  # e3 + e8 = 5
&lt;/syntaxhighlight&gt;
All tests produced a consistent value of &lt;math&gt;R.&lt;/math&gt;

====Checking====
=====Examples of current nodes=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0101network01.png|&lt;small&gt;&lt;math&gt;i_1 + i_{10} = i_2&lt;/math&gt;&lt;/small&gt;
File:0101network02.png|&lt;small&gt;&lt;math&gt;i_6 + i_8 = i_{12}&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}

=====Examples of voltage loops=====

{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0108network00.png|&lt;small&gt;Top and Bottom.&lt;math&gt;e_1 + e_2 - e_4 - e_3 = 0&lt;/math&gt;&lt;math&gt;e_5 + e_6 - e_8 - e_7 = 0&lt;/math&gt;&lt;/small&gt;
File:0108network01.png|&lt;small&gt;Two sides in background.&lt;math&gt;e_5 + e_{10} - e_1 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_6 + e_{12} - e_2 - e_{10} = 0&lt;/math&gt;&lt;/small&gt;
File:0108network02.png|&lt;small&gt;Two sides in foreground.&lt;math&gt;e_7 + e_{11} - e_3 - e_9 = 0&lt;/math&gt;&lt;math&gt;e_8 + e_{12} - e_4 - e_{11} = 0&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}
=====Python code=====
&lt;syntaxhighlight lang=python&gt;
t1 = (
# 3 nodes on top
'i1 + i10 - i2',
'i1 + i3 - i9',
'i3 + i11 - i4',

# 3 nodes on bottom
'i10 + i6 - i5',
'i11 + i8 - i7',
'i6 + i8 - i12',

# Total input current equals total output current.
'i5 + i7 + i9 - i2 - i4 - i12',

# Voltage loops
# top
'e1 + e2 - e4 - e3',

# bottom
'e5 + e6 - e8 - e7',

# 2 sides in background
'e9 + e1 - e10 - e5',
'e10 + e2 - e12 - e6',

# 2 sides in foreground
'e9 + e3 - e11 - e7',
'e11 + e4 - e12 - e8',

# Some elaborate loops.
'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7',
'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5',
)
&lt;/syntaxhighlight&gt;
Result of each of the above calculations should be &lt;math&gt;0.&lt;/math&gt;

Because of small rounding errors, result may not be exactly &lt;math&gt;0.&lt;/math&gt;

See values_of_i and values_of_e above.

This code verifies that result is close enough to &lt;math&gt;0&lt;/math&gt; to be acceptable.

For example:

&lt;math&gt;\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.&lt;/math&gt;

&lt;math&gt;\text{v1} = e_{10} + e_2 + e_{11} + e_5,&lt;/math&gt; sum of positive values.

&lt;math&gt;\text{status = (abs(result)} &lt; \text{(v1}*\text{Tolerance)). status}&lt;/math&gt; should be &lt;math&gt;\text{True.}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

import re

for str1 in t1 :
    str2 = re.sub ('\+', ';+', str1)
    str2 = re.sub ('\-', ';-', str2)
    L1 = str2.split(';')
    positives = ''.join([ v for v in L1 if '-' not in v ])
    result = eval(str1)
    v1 = eval( positives )
    status = (abs(result) &lt; (v1*Tolerance))
    str3 = (str1+(' '*40))[:42]
    str4 = ((' '*10) + str(result))[-10:]
    str5 = '{}{}    {}'.format(str3,str4,status)
    print ( str5 )
&lt;/syntaxhighlight&gt;

{| class=&quot;wikitable&quot;
|- 
! Calculation || Result !! Result Acceptable
|-
| &lt;math&gt; i_1 + i_{10} - i_2 &lt;/math&gt;
| &lt;math&gt; -1.2E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_1 + i_3 - i_9 &lt;/math&gt;
| &lt;math&gt; -6E-29 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_3 + i_{11} - i_4 &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_{10} + i_6 - i_5 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt; 
|-
| &lt;math&gt; i_{11} + i_8 - i_7 &lt;/math&gt;
| &lt;math&gt; 0E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; i_6 + i_8 - i_{12} &lt;/math&gt;
| &lt;math&gt; -2E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} &lt;/math&gt;
| &lt;math&gt; -3E-29 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_1 + e_2 - e_4 - e_3 &lt;/math&gt;
| &lt;math&gt; 0E-28 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_5 + e_6 - e_8 - e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-  
| &lt;math&gt; e_9 + e_1 - e_{10} - e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_6 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt; 
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_9 + e_3 - e_{11} - e_7 &lt;/math&gt;
| &lt;math&gt; -1.3E-26 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{11} + e_4 - e_{12} - e_8 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|-
| &lt;math&gt; e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 &lt;/math&gt;
| &lt;math&gt; 0E-27 &lt;/math&gt;
| &lt;math&gt;\text{True}&lt;/math&gt;
|}
===Big Matrix 64 by 65===
{{RoundBoxTop|theme=2}}
[[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.'''
&lt;/br&gt;
Positive current flow is assumed to be from bottom to top or from left to right.
]]
A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

&lt;syntaxhighlight lang=python&gt;
# python code
values_of_R = (
    R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, 
    R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, 
    R30, R31, R32, R33, R34, R35, R36, R37, R38, R39,
    R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, 
    R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, 
    R60, R61, R62, R63, R64, R65, R66, R67, R68, R69,
    R70, R71, R72, R73, 
) = [ v for v in range (100,731,10) ]
&lt;/syntaxhighlight&gt;
While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64
resistors and 64 simultaneous equations to be solved.
{{RoundBoxBottom}}
====Create matrix====
An examination of the circuit produces the following conditions:

&lt;syntaxhighlight lang=python&gt;
# python code

iconditions = [
              &quot;+ i10 + i70 - i44&quot; ,        &quot;+ i11 - i10 - i49&quot; ,        &quot;+ i12 - i11 - i54&quot;,
              &quot;+ i13 - i12 - i59&quot; ,        &quot;+ i14 - i13 - i64&quot; ,        &quot;+ i15 + i44 - i43&quot;,
  &quot;+ i16 + i49 - i15 - i48 - i70&quot; ,  &quot;+ i17 + i54 - i16 - i53&quot; ,  &quot;+ i18 + i59 - i17 - i58&quot;,
  &quot;+ i19 + i64 + i71 - i18 - i63&quot; ,        &quot;+ i69 - i19 - i68&quot; ,        &quot;+ i20 + i43 - i42&quot;,
        &quot;+ i21 + i48 - i20 - i47&quot; ,  &quot;+ i22 + i53 - i21 - i52&quot; ,  &quot;+ i23 + i58 - i22 - i57&quot;,
        &quot;+ i24 + i63 - i23 - i62&quot; ,        &quot;+ i68 - i24 - i67&quot; ,        &quot;+ i25 + i42 - i41&quot;,
        &quot;+ i26 + i47 - i25 - i46&quot; ,  &quot;+ i27 + i52 - i26 - i51&quot; ,  &quot;+ i28 + i57 - i27 - i56&quot;,
        &quot;+ i29 + i62 - i28 - i61&quot; ,        &quot;+ i67 - i29 - i66&quot; ,        &quot;+ i30 + i41 - i40&quot;,
  &quot;+ i31 + i46 - i30 - i45 - i72&quot; ,  &quot;+ i32 + i51 - i31 - i50&quot; ,  &quot;+ i33 + i56 - i32 - i55&quot;,
  &quot;+ i34 + i61 + i73 - i33 - i60&quot; ,        &quot;+ i66 - i34 - i65&quot; ,        &quot;+ i36 + i45 - i35&quot;,
              &quot;+ i37 + i50 - i36&quot; ,        &quot;+ i38 + i55 - i37&quot; ,        &quot;+ i39 + i60 - i38&quot;,
              &quot;+ i65 - i39 - i73&quot;
]

sx = 'len(iconditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(iconditions) = 34
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
econditions = [
  &quot;+ e40 + e30 - e35 - e45&quot; ,  &quot;+ e41 + e25 - e30 - e46&quot; ,  &quot;+ e42 + e20 - e25 - e47&quot;,
  &quot;+ e43 + e15 - e20 - e48&quot; ,  &quot;+ e44 + e10 - e15 - e49&quot; ,  &quot;+ e45 + e31 - e36 - e50&quot;,
  &quot;+ e46 + e26 - e31 - e51&quot; ,  &quot;+ e47 + e21 - e26 - e52&quot; ,  &quot;+ e48 + e16 - e21 - e53&quot;,
  &quot;+ e49 + e11 - e16 - e54&quot; ,  &quot;+ e50 + e32 - e37 - e55&quot; ,  &quot;+ e51 + e27 - e32 - e56&quot;,
  &quot;+ e52 + e22 - e27 - e57&quot; ,  &quot;+ e53 + e17 - e22 - e58&quot; ,  &quot;+ e54 + e12 - e17 - e59&quot;,
  &quot;+ e55 + e33 - e38 - e60&quot; ,  &quot;+ e56 + e28 - e33 - e61&quot; ,  &quot;+ e57 + e23 - e28 - e62&quot;,
  &quot;+ e58 + e18 - e23 - e63&quot; ,  &quot;+ e59 + e13 - e18 - e64&quot; ,  &quot;+ e60 + e34 - e39 - e65&quot;,
  &quot;+ e61 + e29 - e34 - e66&quot; ,  &quot;+ e62 + e24 - e29 - e67&quot; ,  &quot;+ e63 + e19 - e24 - e68&quot;,
  &quot;+ e64 + e14 - e19 - e69&quot; ,        &quot;+ e44 + e70 - e15&quot; ,        &quot;+ e64 + e14 - e71&quot;,
        &quot;+ e40 + e30 - e72&quot; ,        &quot;+ e60 + e73 - e39&quot;
]

sx = 'len(econditions)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(econditions) = 29
&lt;/syntaxhighlight&gt;
In iconditions and econditions there are 63 conditions that include all values of i and e.
These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is:
&lt;syntaxhighlight lang=python&gt;
matrix = [
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,],
  [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,],
  [0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,],
  [0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,-350,0,0,0,0,400,0,0,0,0,-450,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,250,0,0,0,0,-300,0,0,0,0,0,0,0,0,0,0,410,0,0,0,0,-460,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,200,0,0,0,0,-250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,420,0,0,0,0,-470,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,150,0,0,0,0,-200,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,430,0,0,0,0,-480,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [100,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,-490,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,310,0,0,0,0,-360,0,0,0,0,0,0,0,0,450,0,0,0,0,-500,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,260,0,0,0,0,-310,0,0,0,0,0,0,0,0,0,0,0,0,0,0,460,0,0,0,0,-510,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,210,0,0,0,0,-260,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,470,0,0,0,0,-520,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,160,0,0,0,0,-210,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,480,0,0,0,0,-530,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,110,0,0,0,0,-160,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,490,0,0,0,0,-540,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,320,0,0,0,0,-370,0,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,-550,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,270,0,0,0,0,-320,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,510,0,0,0,0,-560,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,220,0,0,0,0,-270,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,520,0,0,0,0,-570,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,170,0,0,0,0,-220,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,530,0,0,0,0,-580,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,120,0,0,0,0,-170,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,540,0,0,0,0,-590,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,-380,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,550,0,0,0,0,-600,0,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,280,0,0,0,0,-330,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,560,0,0,0,0,-610,0,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,230,0,0,0,0,-280,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,570,0,0,0,0,-620,0,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,180,0,0,0,0,-230,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,580,0,0,0,0,-630,0,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,130,0,0,0,0,-180,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,590,0,0,0,0,-640,0,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,340,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,-650,0,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,290,0,0,0,0,-340,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,610,0,0,0,0,-660,0,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,-290,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620,0,0,0,0,-670,0,0,0,0,0,0,0,],
  [0,0,0,0,0,0,0,0,0,190,0,0,0,0,-240,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,630,0,0,0,0,-680,0,0,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,],
  [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,],
  [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,],
  [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,],
  [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,],
]

sx = 'len(matrix)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(matrix) = 64
&lt;/syntaxhighlight&gt;
Bottom line of matrix is equivalent to: &lt;math&gt;(1)i10 + (-1) = 0&lt;/math&gt; or &lt;math&gt;i10 = 1.&lt;/math&gt;
====First result====
The following conditions are included for testing:
&lt;syntaxhighlight lang=python&gt;
more_econditions = [
  &quot;+ e49 + e70 - e10&quot;,
  &quot;+ e19 + e69 - e71&quot;,
  &quot;+ e35 + e45 - e72&quot;,
  &quot;+ e65 + e73 - e34&quot;,
] 

extra_test_conditions = [
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49&quot;,
  &quot;+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54&quot;,
  &quot;+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59&quot;,
  &quot;+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64&quot;,
  &quot;+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19&quot;,
  &quot;+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24&quot;,
  &quot;+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29&quot;,
  &quot;+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39&quot;,
  &quot;+ i35 + i40 + i72 - i14 - i69 - i71&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69&quot;,
  &quot;+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14&quot;,
  &quot;+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14&quot;,
  &quot;+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14&quot;,
  &quot;+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14&quot;
]
&lt;/syntaxhighlight&gt;
Produce results:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    import decimal
    dD = decimal.Decimal
    dgt = decimal.getcontext()
    Precision = dgt.prec = 60                   # Adjust as necessary.
    Tolerance = dD(&quot;1e-&quot; + str(Precision-4))    # Adjust as necessary.
    
    import re # Regular expressions.

def produce_result (input, flag = 0) :
    thisName = 'produce_result (input, flag = {}) :'.format(flag)

    values_of_i = solveMbyN (input)

    (   i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, 
        i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, 
        i30, i31, i32, i33, i34, i35, i36, i37, i38, i39,
        i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, 
        i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, 
        i60, i61, i62, i63, i64, i65, i66, i67, i68, i69,
        i70, i71, i72, i73, 
    ) = values_of_i

    values_of_e = (
        e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, 
        e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, 
        e30, e31, e32, e33, e34, e35, e36, e37, e38, e39,
        e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, 
        e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, 
        e60, e61, e62, e63, e64, e65, e66, e67, e68, e69,
        e70, e71, e72, e73, 
    ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ]

    E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14'
    E1 = eval(E1_)
    I1 = eval('i40 + i72 + i35')
    R1 = E1/I1

    if not flag : return R1
    # Check all conditions.
    # compiled regular expression plus, compiled regular expression minus
    crep,crem = [ re.compile(re.escape(v)) for v in '+-' ]
    # 
    # This code verifies that result of evaluation of condition 
    # is close enough to 0 to be called 0.
    # 
    for condition in (iconditions + econditions + more_econditions + extra_test_conditions):
        str2 = crep.sub(';+', condition)
        str3 = crem.sub(';-', str2)
        positives = [ v for v in str3.split(';') if ('+' in v) ]
        pos_sum = eval(''.join(positives))
        sum = eval(condition)
        # pos_sum - neg_sum = sum
        neg_sum = pos_sum - sum
        min,max = sorted((neg_sum, pos_sum))
        status = (abs(sum) &lt;= (Tolerance*min))
        if not status :
            print (thisName, 'condition failed:')
            print ('   ', condition, sum)
            return None

    return R1

# Convert all values in matrix to Decimal objects
matrix = [ [ dD(str(eval(str(p)))) for p in v  ] for v in matrix ]

enable_reduceRow = 1

R = produce_result (matrix,1) 

sx = 'R' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
R = 723.928575988442547699732563335265663515838075102937503680598
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 60.
====Matrix corrected====
{{RoundBoxTop|theme=2}}
[[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors'''
&lt;/br&gt;
Diagram has been corrected to show positive direction of current through R70, R73.
]]
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right.

{{RoundBoxBottom}}

&lt;syntaxhighlight lang=python&gt;
number_of_columns = 65
output = []
# 
# Test this matrix 64 times using 
# i10 = 1
# i11 = 1.1
# i12 = 1.2
# .......
# .......
# i70 = -7
# i71 = 7.1
# i72 = 7.2
# i73 = -7.3
# 
for I in range(10, 74):
    new_line = [ dD(0) ] * number_of_columns
    place_in_line = I - 10
    new_line[place_in_line] = dD(1)
    i = dD(I)/10
    if I in (70,73) : new_line[-1] = i
    else : new_line[-1] = -i
    matrix[-1] = new_line
    r = produce_result (matrix,1)
    if r : output += [ r ]
    
sx = 'len(output)' ; print (sx, '=', eval(sx))

# Produce results with slightly reduced precision:

dgt.prec -= 4
set2 = { r+0 for r in output }
dgt.prec += 4
    
len2 = len(set2)
if len2 == 1 :
    r, = set2
    print ('R =', r)
else :
    print (len2, 'values of R:')
    for v in set2 : print ('   ', v)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(output) = 64 
R = 723.92857598844254769973256333526566351583807510293750368
&lt;/syntaxhighlight&gt;
&lt;math&gt;\text{R}&lt;/math&gt; is printed with precision of 56.

With precision of 56 all 64 results are equal.

[[Category:Equations]]</text>
      <sha1>lelbi14kpldo3ii2u0v7j4qdpvpenq0</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Assessment/Chapter/Readability</title>
    <ns>0</ns>
    <id>278794</id>
    <revision>
      <id>2834808</id>
      <parentid>2676664</parentid>
      <timestamp>2026-09-28T06:17:35Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Add table of readability statistics based on ChatGPT: https://chatgpt.com/share/6aba066c-b280-83ec-8a11-f6a48023ca19</comment>
      <origin>2834808</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2963" sha1="bw6x00p4ctpdz3hlde3ulbrwwwo28li" xml:space="preserve">{{title|Readability}}

One way to improve a chapter draft is to conduct a [[w:Readability|readability]] analysis.

Most readability analyses use formulas based on a combination of:
* length of sentences (shorter is better)
* length of words (shorter is better)

==Readability targets==
The most common readability statistics are shown in Table 1.

'''Table 1.'''

''Readability Statistics and Recommended Ranges''

{| class=&quot;wikitable&quot;
! Rank
! Indicator
! What does it mean?
! Desirable range for university student writing
|-
| 1
| '''Flesch Reading Ease'''
| Estimates how easy the text is to read. Higher scores mean easier reading. The score is based mainly on sentence length and word length.
| '''50–70''' is a useful target. Scores above 70 are easier to read; scores below 50 indicate more difficult text.
|-
| 2
| '''Flesch–Kincaid Grade Level'''
| Estimates the US school grade level needed to understand the text. Higher scores indicate more difficult text.
| '''8–12''' is a useful target for general university writing. Higher scores may be appropriate when technical or specialised language is necessary.
|-
| 3
| '''Gunning Fog Index'''
| Estimates the years of formal education needed to understand the text. It gives considerable weight to long words and long sentences.
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 4
| '''SMOG'''
| Estimates the years of education needed to understand the text, based particularly on the number of longer words (polysyllabic words).
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 5
| '''Automated Readability Index (ARI)'''
| Estimates the US grade level needed to understand the text, based mainly on word and sentence length.
| '''8–12''' is a useful target for general university writing. Higher scores indicate more difficult text.
|}

==Readability tools==
Paste the URL into the [https://www.webfx.com/tools/read-able/website-readability.html webfx readability tool]. Aim for a readability score of at least 50. For example, [[Motivation and emotion/Book/2020/Funerals and grief work|funerals and grief work]] (2020) gets a score of 57.

For more diagnostic detail, paste text into the [https://hemingwayapp.com/ Hemingway app] or asking a [[Motivation and emotion/Assessment/Using generative AI|genAI tool]] for a readability analysis.

Can you suggest other readability tools?

==See also==
* [[w:Readability|Readability]] (Wikipedia)
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI guidelines]]

==External links==
* [https://hemingwayapp.com/ Hemingway app] (hemingway.app)
* [https://toolsinpocket.com/tools/readability-checker/ Readability checker] (toolsinpocket.com)
* [https://www.webfx.com/tools/read-able/website-readability.html Webfx readability tool]

[[Category:How to write]]
[[Category:Motivation and emotion/Assessment/Chapter]]</text>
      <sha1>bw6x00p4ctpdz3hlde3ulbrwwwo28li</sha1>
    </revision>
    <revision>
      <id>2834809</id>
      <parentid>2834808</parentid>
      <timestamp>2026-09-28T06:18:20Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2834809</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2997" sha1="1l4ty07kwt1fyhzu37ncx62wgh3jv8n" xml:space="preserve">{{title|Readability}}

One way to improve a chapter draft is to conduct a [[w:Readability|readability]] analysis.

Most readability analyses use formulas based on a combination of:
* length of sentences (shorter sentences are easier to read)
* length of words (shorter words are easier to read)

==Readability targets==
The most common readability statistics are shown in Table 1.

'''Table 1.'''

''Readability Statistics and Recommended Ranges''

{| class=&quot;wikitable&quot;
! Rank
! Indicator
! What does it mean?
! Desirable range for university student writing
|-
| 1
| '''Flesch Reading Ease'''
| Estimates how easy the text is to read. Higher scores mean easier reading. The score is based mainly on sentence length and word length.
| '''50–70''' is a useful target. Scores above 70 are easier to read; scores below 50 indicate more difficult text.
|-
| 2
| '''Flesch–Kincaid Grade Level'''
| Estimates the US school grade level needed to understand the text. Higher scores indicate more difficult text.
| '''8–12''' is a useful target for general university writing. Higher scores may be appropriate when technical or specialised language is necessary.
|-
| 3
| '''Gunning Fog Index'''
| Estimates the years of formal education needed to understand the text. It gives considerable weight to long words and long sentences.
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 4
| '''SMOG'''
| Estimates the years of education needed to understand the text, based particularly on the number of longer words (polysyllabic words).
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 5
| '''Automated Readability Index (ARI)'''
| Estimates the US grade level needed to understand the text, based mainly on word and sentence length.
| '''8–12''' is a useful target for general university writing. Higher scores indicate more difficult text.
|}

==Readability tools==
Paste the URL into the [https://www.webfx.com/tools/read-able/website-readability.html webfx readability tool]. Aim for a readability score of at least 50. For example, [[Motivation and emotion/Book/2020/Funerals and grief work|funerals and grief work]] (2020) gets a score of 57.

For more diagnostic detail, paste text into the [https://hemingwayapp.com/ Hemingway app] or asking a [[Motivation and emotion/Assessment/Using generative AI|genAI tool]] for a readability analysis.

Can you suggest other readability tools?

==See also==
* [[w:Readability|Readability]] (Wikipedia)
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI guidelines]]

==External links==
* [https://hemingwayapp.com/ Hemingway app] (hemingway.app)
* [https://toolsinpocket.com/tools/readability-checker/ Readability checker] (toolsinpocket.com)
* [https://www.webfx.com/tools/read-able/website-readability.html Webfx readability tool]

[[Category:How to write]]
[[Category:Motivation and emotion/Assessment/Chapter]]</text>
      <sha1>1l4ty07kwt1fyhzu37ncx62wgh3jv8n</sha1>
    </revision>
    <revision>
      <id>2834810</id>
      <parentid>2834809</parentid>
      <timestamp>2026-09-28T06:19:41Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Readability tools */</comment>
      <origin>2834810</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3042" sha1="ntmo1qh3d3swiv9o7ws0nc0720ol3s0" xml:space="preserve">{{title|Readability}}

One way to improve a chapter draft is to conduct a [[w:Readability|readability]] analysis.

Most readability analyses use formulas based on a combination of:
* length of sentences (shorter sentences are easier to read)
* length of words (shorter words are easier to read)

==Readability targets==
The most common readability statistics are shown in Table 1.

'''Table 1.'''

''Readability Statistics and Recommended Ranges''

{| class=&quot;wikitable&quot;
! Rank
! Indicator
! What does it mean?
! Desirable range for university student writing
|-
| 1
| '''Flesch Reading Ease'''
| Estimates how easy the text is to read. Higher scores mean easier reading. The score is based mainly on sentence length and word length.
| '''50–70''' is a useful target. Scores above 70 are easier to read; scores below 50 indicate more difficult text.
|-
| 2
| '''Flesch–Kincaid Grade Level'''
| Estimates the US school grade level needed to understand the text. Higher scores indicate more difficult text.
| '''8–12''' is a useful target for general university writing. Higher scores may be appropriate when technical or specialised language is necessary.
|-
| 3
| '''Gunning Fog Index'''
| Estimates the years of formal education needed to understand the text. It gives considerable weight to long words and long sentences.
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 4
| '''SMOG'''
| Estimates the years of education needed to understand the text, based particularly on the number of longer words (polysyllabic words).
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 5
| '''Automated Readability Index (ARI)'''
| Estimates the US grade level needed to understand the text, based mainly on word and sentence length.
| '''8–12''' is a useful target for general university writing. Higher scores indicate more difficult text.
|}

==Readability tools==
Paste the [[Motivation and emotion/Book|book chapter]] URL into the [https://www.webfx.com/tools/read-able/website-readability.html webfx readability tool]. Aim for a readability score of at least 50. For example, [[Motivation and emotion/Book/2020/Funerals and grief work|funerals and grief work]] (2020) gets a score of 57.

For more diagnostic detail, paste text into the [https://hemingwayapp.com/ Hemingway app] or asking a [[Motivation and emotion/Assessment/Using generative AI|genAI tool]] for a readability analysis.

Can you suggest other readability tools?

==See also==
* [[w:Readability|Readability]] (Wikipedia)
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI guidelines]]

==External links==
* [https://hemingwayapp.com/ Hemingway app] (hemingway.app)
* [https://toolsinpocket.com/tools/readability-checker/ Readability checker] (toolsinpocket.com)
* [https://www.webfx.com/tools/read-able/website-readability.html Webfx readability tool]

[[Category:How to write]]
[[Category:Motivation and emotion/Assessment/Chapter]]</text>
      <sha1>ntmo1qh3d3swiv9o7ws0nc0720ol3s0</sha1>
    </revision>
    <revision>
      <id>2834811</id>
      <parentid>2834810</parentid>
      <timestamp>2026-09-28T06:25:49Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Readability tools */ Add sub-headings</comment>
      <origin>2834811</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3239" sha1="8fqqs4hfhpcfm2sc6pv8sh89ilh05s2" xml:space="preserve">{{title|Readability}}

One way to improve a chapter draft is to conduct a [[w:Readability|readability]] analysis.

Most readability analyses use formulas based on a combination of:
* length of sentences (shorter sentences are easier to read)
* length of words (shorter words are easier to read)

==Readability targets==
The most common readability statistics are shown in Table 1.

'''Table 1.'''

''Readability Statistics and Recommended Ranges''

{| class=&quot;wikitable&quot;
! Rank
! Indicator
! What does it mean?
! Desirable range for university student writing
|-
| 1
| '''Flesch Reading Ease'''
| Estimates how easy the text is to read. Higher scores mean easier reading. The score is based mainly on sentence length and word length.
| '''50–70''' is a useful target. Scores above 70 are easier to read; scores below 50 indicate more difficult text.
|-
| 2
| '''Flesch–Kincaid Grade Level'''
| Estimates the US school grade level needed to understand the text. Higher scores indicate more difficult text.
| '''8–12''' is a useful target for general university writing. Higher scores may be appropriate when technical or specialised language is necessary.
|-
| 3
| '''Gunning Fog Index'''
| Estimates the years of formal education needed to understand the text. It gives considerable weight to long words and long sentences.
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 4
| '''SMOG'''
| Estimates the years of education needed to understand the text, based particularly on the number of longer words (polysyllabic words).
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 5
| '''Automated Readability Index (ARI)'''
| Estimates the US grade level needed to understand the text, based mainly on word and sentence length.
| '''8–12''' is a useful target for general university writing. Higher scores indicate more difficult text.
|}

==Readability tools==
This section recommends readability analysis tools, starting from simple to more complex.

===Webfx===
* Paste the [[Motivation and emotion/Book|book chapter]] text into the [https://www.webfx.com/tools/read-able/website-readability.html webfx readability tool]
* Calculate readability
* Aim for a readability score of at least 50. For example, [[Motivation and emotion/Book/2020/Funerals and grief work|funerals and grief work]] (2020) gets a Flesch Kincaid Reading Ease score of 57.

===Hemingway===
* For more diagnostic detail, paste text into the [https://hemingwayapp.com/ Hemingway app]

===GenAI===
* Ask a [[Motivation and emotion/Assessment/Using generative AI|genAI tool]] for a readability analysis

===Other===
Can you suggest other readability tools?

==See also==
* [[w:Readability|Readability]] (Wikipedia)
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI guidelines]]

==External links==
* [https://hemingwayapp.com/ Hemingway app] (hemingway.app)
* [https://toolsinpocket.com/tools/readability-checker/ Readability checker] (toolsinpocket.com)
* [https://www.webfx.com/tools/read-able/website-readability.html Webfx readability tool]

[[Category:How to write]]
[[Category:Motivation and emotion/Assessment/Chapter]]</text>
      <sha1>8fqqs4hfhpcfm2sc6pv8sh89ilh05s2</sha1>
    </revision>
    <revision>
      <id>2834813</id>
      <parentid>2834811</parentid>
      <timestamp>2026-09-28T06:30:06Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Readability tools */ * The key is to use readability analysis to help simplify a chapter's language in order to appeal and be accessible to a wide audience</comment>
      <origin>2834813</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3374" sha1="ad968jljdvpby35jsl27g0b3k8tuvwp" xml:space="preserve">{{title|Readability}}

One way to improve a chapter draft is to conduct a [[w:Readability|readability]] analysis.

Most readability analyses use formulas based on a combination of:
* length of sentences (shorter sentences are easier to read)
* length of words (shorter words are easier to read)

==Readability targets==
The most common readability statistics are shown in Table 1.

'''Table 1.'''

''Readability Statistics and Recommended Ranges''

{| class=&quot;wikitable&quot;
! Rank
! Indicator
! What does it mean?
! Desirable range for university student writing
|-
| 1
| '''Flesch Reading Ease'''
| Estimates how easy the text is to read. Higher scores mean easier reading. The score is based mainly on sentence length and word length.
| '''50–70''' is a useful target. Scores above 70 are easier to read; scores below 50 indicate more difficult text.
|-
| 2
| '''Flesch–Kincaid Grade Level'''
| Estimates the US school grade level needed to understand the text. Higher scores indicate more difficult text.
| '''8–12''' is a useful target for general university writing. Higher scores may be appropriate when technical or specialised language is necessary.
|-
| 3
| '''Gunning Fog Index'''
| Estimates the years of formal education needed to understand the text. It gives considerable weight to long words and long sentences.
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 4
| '''SMOG'''
| Estimates the years of education needed to understand the text, based particularly on the number of longer words (polysyllabic words).
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 5
| '''Automated Readability Index (ARI)'''
| Estimates the US grade level needed to understand the text, based mainly on word and sentence length.
| '''8–12''' is a useful target for general university writing. Higher scores indicate more difficult text.
|}

==Readability tools==
* This section recommends readability analysis tools, ranging from simple to more complex
* The key is to use readability analysis to help simplify a chapter's language in order to appeal and be accessible to a wide audience

===Webfx===
* Paste the [[Motivation and emotion/Book|book chapter]] text into the [https://www.webfx.com/tools/read-able/website-readability.html webfx readability tool]
* Calculate readability
* Aim for a readability score of at least 50. For example, [[Motivation and emotion/Book/2020/Funerals and grief work|funerals and grief work]] (2020) gets a Flesch Kincaid Reading Ease score of 57.

===Hemingway===
* For more diagnostic detail, paste text into the [https://hemingwayapp.com/ Hemingway app]

===GenAI===
* Ask a [[Motivation and emotion/Assessment/Using generative AI|genAI tool]] for a readability analysis

===Other===
Can you suggest other readability tools?

==See also==
* [[w:Readability|Readability]] (Wikipedia)
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI guidelines]]

==External links==
* [https://hemingwayapp.com/ Hemingway app] (hemingway.app)
* [https://toolsinpocket.com/tools/readability-checker/ Readability checker] (toolsinpocket.com)
* [https://www.webfx.com/tools/read-able/website-readability.html Webfx readability tool]

[[Category:How to write]]
[[Category:Motivation and emotion/Assessment/Chapter]]</text>
      <sha1>ad968jljdvpby35jsl27g0b3k8tuvwp</sha1>
    </revision>
    <revision>
      <id>2834814</id>
      <parentid>2834813</parentid>
      <timestamp>2026-09-28T06:31:37Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2834814</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3383" sha1="3tmyp9y0zu3ykuhjy3gqzv76gpd2kav" xml:space="preserve">{{title|Readability analysis}}

One way to improve a chapter draft is to conduct a [[w:Readability|readability]] analysis.

Most readability analyses use formulas based on a combination of:
* length of sentences (shorter sentences are easier to read)
* length of words (shorter words are easier to read)

==Readability targets==
The most common readability statistics are shown in Table 1.

'''Table 1.'''

''Readability Statistics and Recommended Ranges''

{| class=&quot;wikitable&quot;
! Rank
! Indicator
! What does it mean?
! Desirable range for university student writing
|-
| 1
| '''Flesch Reading Ease'''
| Estimates how easy the text is to read. Higher scores mean easier reading. The score is based mainly on sentence length and word length.
| '''50–70''' is a useful target. Scores above 70 are easier to read; scores below 50 indicate more difficult text.
|-
| 2
| '''Flesch–Kincaid Grade Level'''
| Estimates the US school grade level needed to understand the text. Higher scores indicate more difficult text.
| '''8–12''' is a useful target for general university writing. Higher scores may be appropriate when technical or specialised language is necessary.
|-
| 3
| '''Gunning Fog Index'''
| Estimates the years of formal education needed to understand the text. It gives considerable weight to long words and long sentences.
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 4
| '''SMOG'''
| Estimates the years of education needed to understand the text, based particularly on the number of longer words (polysyllabic words).
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 5
| '''Automated Readability Index (ARI)'''
| Estimates the US grade level needed to understand the text, based mainly on word and sentence length.
| '''8–12''' is a useful target for general university writing. Higher scores indicate more difficult text.
|}

==Readability tools==
* This section recommends readability analysis tools, ranging from simple to more complex
* The key is to use readability analysis to help simplify a chapter's language in order to appeal and be accessible to a wide audience

===Webfx===
* Paste the [[Motivation and emotion/Book|book chapter]] text into the [https://www.webfx.com/tools/read-able/website-readability.html webfx readability tool]
* Calculate readability
* Aim for a readability score of at least 50. For example, [[Motivation and emotion/Book/2020/Funerals and grief work|funerals and grief work]] (2020) gets a Flesch Kincaid Reading Ease score of 57.

===Hemingway===
* For more diagnostic detail, paste text into the [https://hemingwayapp.com/ Hemingway app]

===GenAI===
* Ask a [[Motivation and emotion/Assessment/Using generative AI|genAI tool]] for a readability analysis

===Other===
Can you suggest other readability tools?

==See also==
* [[w:Readability|Readability]] (Wikipedia)
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI guidelines]]

==External links==
* [https://hemingwayapp.com/ Hemingway app] (hemingway.app)
* [https://toolsinpocket.com/tools/readability-checker/ Readability checker] (toolsinpocket.com)
* [https://www.webfx.com/tools/read-able/website-readability.html Webfx readability tool]

[[Category:How to write]]
[[Category:Motivation and emotion/Assessment/Chapter]]</text>
      <sha1>3tmyp9y0zu3ykuhjy3gqzv76gpd2kav</sha1>
    </revision>
    <revision>
      <id>2834815</id>
      <parentid>2834814</parentid>
      <timestamp>2026-09-28T06:34:21Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Shorter words and sentences are more readable. Therefore,</comment>
      <origin>2834815</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3479" sha1="hi2cwkjzk4xt1201fo519uicv38ftp6" xml:space="preserve">{{title|Readability analysis}}

One way to improve a [[Motivation and emotion/Assessment/Chapter|book chapter]] is to conduct a [[w:Readability|readability]] analysis.

Shorter words and sentences are more readable. Therefore, most readability statistics are based on a combination of:
* length of sentences (shorter sentences are easier to read)
* length of words (shorter words are easier to read)

==Readability targets==
The most common readability statistics are shown in Table 1.

'''Table 1.'''

''Readability Statistics and Recommended Ranges''

{| class=&quot;wikitable&quot;
! Rank
! Indicator
! What does it mean?
! Desirable range for university student writing
|-
| 1
| '''Flesch Reading Ease'''
| Estimates how easy the text is to read. Higher scores mean easier reading. The score is based mainly on sentence length and word length.
| '''50–70''' is a useful target. Scores above 70 are easier to read; scores below 50 indicate more difficult text.
|-
| 2
| '''Flesch–Kincaid Grade Level'''
| Estimates the US school grade level needed to understand the text. Higher scores indicate more difficult text.
| '''8–12''' is a useful target for general university writing. Higher scores may be appropriate when technical or specialised language is necessary.
|-
| 3
| '''Gunning Fog Index'''
| Estimates the years of formal education needed to understand the text. It gives considerable weight to long words and long sentences.
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 4
| '''SMOG'''
| Estimates the years of education needed to understand the text, based particularly on the number of longer words (polysyllabic words).
| '''10–14''' is a useful target for university writing. Higher scores indicate more complex text.
|-
| 5
| '''Automated Readability Index (ARI)'''
| Estimates the US grade level needed to understand the text, based mainly on word and sentence length.
| '''8–12''' is a useful target for general university writing. Higher scores indicate more difficult text.
|}

==Readability tools==
* This section recommends readability analysis tools, ranging from simple to more complex
* The key is to use readability analysis to help simplify a chapter's language in order to appeal and be accessible to a wide audience

===Webfx===
* Paste the [[Motivation and emotion/Book|book chapter]] text into the [https://www.webfx.com/tools/read-able/website-readability.html webfx readability tool]
* Calculate readability
* Aim for a readability score of at least 50. For example, [[Motivation and emotion/Book/2020/Funerals and grief work|funerals and grief work]] (2020) gets a Flesch Kincaid Reading Ease score of 57.

===Hemingway===
* For more diagnostic detail, paste text into the [https://hemingwayapp.com/ Hemingway app]

===GenAI===
* Ask a [[Motivation and emotion/Assessment/Using generative AI|genAI tool]] for a readability analysis

===Other===
Can you suggest other readability tools?

==See also==
* [[w:Readability|Readability]] (Wikipedia)
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI guidelines]]

==External links==
* [https://hemingwayapp.com/ Hemingway app] (hemingway.app)
* [https://toolsinpocket.com/tools/readability-checker/ Readability checker] (toolsinpocket.com)
* [https://www.webfx.com/tools/read-able/website-readability.html Webfx readability tool]

[[Category:How to write]]
[[Category:Motivation and emotion/Assessment/Chapter]]</text>
      <sha1>hi2cwkjzk4xt1201fo519uicv38ftp6</sha1>
    </revision>
  </page>
  <page>
    <title>Helping Give Away Psychological Science/Standard Operating Procedures/Speaker Series and Continuing Education</title>
    <ns>0</ns>
    <id>282686</id>
    <revision>
      <id>2834804</id>
      <parentid>2832025</parentid>
      <timestamp>2026-09-28T05:17:14Z</timestamp>
      <contributor>
        <username>~2026-51980-30</username>
        <id>3111742</id>
      </contributor>
      <comment>rv spam</comment>
      <origin>2834804</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15197" sha1="he89cezvwetqqucf13n9wycxg7ojo7n" xml:space="preserve">HGAPS hosts a speaker series every semester, featuring noted experts in the field of clinical psychology and other related fields. These talks are typically for education and intellectual enrichment, and are offered to UNC-Chapel Hill students as well as the general public. Some of these presentations will be targeted for career professionals, offering continuing education (CE) credits for licensure maintenance. Completed series are then added to HGAPS’s online presence through [[Helping Give Away Psychological Science|Wikiversity]] and [https://www.youtube.com/c/HGAPSHelpingGiveAwayPsychologicalScience/videos YouTube]. This SOP will describe the procedure for planning and executing a speaker series, as well as contain notes for securing and distributing CE credit for certain talks.

= Speaker Planning =
The UNC-Chapel Hill chapter of HGAPS has secured funding for the speaker series through UNC’s Student Government Association (SGA). Planning begins over the summer by identifying a number of potential speakers, budgeting for the talks, and adding the series into the larger HGAPS funding application. Other HGAPS chapters hosting speakers will likely use a similar process and timeline. 

== Pre-Semester Planning ==

=== Initial planning ===
* identify potential speakers, semester speaker series topics/themes, and other priorities for the series
* Get approval on speakers from Dr. E. Youngstrom (or local chapter advisor) and the HGAPS Board

=== Funding ===
* You must fill out the grant proposal to be submitted to the student senate and create a funding presentation to be presented to the student senate 
* [[Helping Give Away Psychological Science/Standard Operating Procedures/ Officer Roles/ Treasurer Instructable|SOP for SGA funding]]

=== Contact Speaker(s) ===
* Should be done 2-3 months prior, ideally in the prior term or during the summer 
* Confirm: 
** Date &amp; Time
** Length
** In-person v. Zoom or other videoconferencing platform
** General topic of presentation
** Whether presentation will be for continuing education credit
* If the funding request is still under review by SGA, discuss with the Executive Director and/or Faculty Advisor about how to address potential honorarium available. May have to ask for tentative commitments, or commitments independent of whether an honorarium can be given.
* Ask if speaker needs assistance with travel arrangements (i.e., hotel, flights, ground transportation)

=== HGAPS Google Drive and Organization ===
Create new [https://drive.google.com/drive/u/6/folders/1mwuXJeysH13XeMAXSjpaChKtJnVGhFVg drive folder] for each speaker using the [https://drive.google.com/drive/folders/11h0zzeeEZ_zbkJlUD3vzc3B6HFobQvkb?usp=sharing speaker kit], which contains blank revisions of the forms below:
* &lt;u&gt;IRS W9 Form&lt;/u&gt; - government tax information form, asking for social security number and other financial information for HGAPS tax filing. Must receive from every speaker receiving any form of payment from HGAPS.
* &lt;u&gt;HGAPS Speaker Series Cx Form&lt;/u&gt; - contract for services with HGAPS as an independent contractor. Lists expectations, pay, and basic information about the presentation to be given. Will also contain a Speaker Release Form. 
* &lt;u&gt;HGAPS Speaker Release Form&lt;/u&gt; - media waiver, allowing HGAPS to use recordings of presentations without needing permission or approval from the speaker going forward.
* &lt;u&gt;HGAPS Non-CE Speaker Planning Sheet (Non-CE)&lt;/u&gt; - place to collect information about the speaker, for internal use. Don’t necessarily need to fill in each item. 
* ''&lt;u&gt;NCPA Co-Sponsorship Application&lt;/u&gt; - application form to secure CE credit for speaker event, used by NCPA (see below). Speaker Kit also contains HGAPS IRS approval letter as an attachment for this application form''.
Collect following information from the speaker:
* Speaker bio
* Speaker photo/image
* Speaker CV
* Presentation title and description
* 3-4 Learning Objectives for the presentation
* Preferred zoom if hybrid or online: this could be their personal zoom, the Treasurer or President's zoom, or Dr. Youngstrom's zoom.

=== Speaker Financial Information ===
* Send speaker an Independent Contractor Services Form (pre-filled), Speaker Release form, and IRS W9 form
** '''&lt;u&gt;Note&lt;/u&gt;''': W9 &lt;u&gt;should not&lt;/u&gt; be sent back over email - these contain social security numbers, which are sensitive information. Encourage secure file transfer, secure submission software (i.e. Quickbook, or a different secure file transfer link), or traditional mail where appropriate.

=== Social Media and Advertisement ===
* The Social Media Team should should begin broadcasting as soon as details are finalized (date/time, lineup, location/remote session link)
* ''For continuing education'':
** Advertisements should not begin until NCPA co-sponsorship is confirmed
** Advertisements should mention NCPA and availability of CE credit and cost (or that CE is free)
** Social media posts for CE talks should include the objectives and purpose of the talk

== Pre-Presentation Planning ==
Identify someone to introduce the speaker at the beginning of the talk
* Ensure SGA is mentioned during talk, and that the UNC-NC logo appears somewhere in the presentation

=== Social Media and Advertisement ===
* Approximately 2 weeks in advance all of the following advertising material should be created and ready to send out or publish:
** Canva poster, following either previous formats or using an original format. It should include: date and time of the event, location (zoom, hybrid, or in-person), speaker's picture with a sentence or two about the speaker (their position or anything to establish credibility), and the talk title with a sentence about the talk. Add in that this is for CLE credit if CLE credit has been confirmed--if it hasn't, we often obtain it anyway, so it's a safe bet to include.
** HeelLife should include the following and can be created by Social Media, the President, Treasurer or Secretary:
*** Title of the event: previously done in the format of &quot;HGAPS Speaker Series: Talk with Dr. Mandy Jensen-Doss!&quot; but other formats are ok.
*** Brief info, here's a template from past events but feel free to use another method:
**** '''Title:''' Title of the presentation
**** '''With who?''' Brief description of the Speaker to establish credibility. Don't use whole bio, just use their most recent position or what they are known for. Best to say &quot;Professor at University specializing in this type of research&quot;.
**** '''Who is this for?''' Highlight that it's open for everyone. If it's a specific topic, like measurement-based care, note usefulness for graduate students. Perhaps add how it's not specific to Psychology or Neuroscience majors.
**** '''What is it about?''' Copy paste talk description from the professional. This can be substituted with learning outcomes.
**** '''Benefits:''' Coincides with who it is for, but here you can list that it's a great networking opportunity for any undergraduates as well.
**** Additional Information (separate from above with underscores): 
***** '''A little more about Professional:''' Add what remains of their speaker bio excluding the previous description.
*** Theme: typically we choose &quot;Learning&quot;
*** Categories: can include Lecture, Career Exploration &amp; Professional Development, or others such as Health and Well Being. Use best judgement.
*** Perks: select Free Food if it's in person or hybrid!
*** Organization RSVPing: Allow respondents to represent an organization but don't make it required. This hasn't been used much in the past, but there's no harm since it can increase networking.
*** Event evaluation questions: These are optional, but speakers typically enjoy hearing feedback on the presentation. Questions include:
****
*** Photo preview: Use the Canva poster and duplicate it. Then, get rid of their bio info and just leave event information, it should come out to about half a page and fit the HeelLife photo requirements. 
*** Campus Life Experience (CLE) credit:
**** Description: UNC's graduation requirements as of Fall 2023 require incoming and transfer students to attend at least two Campus Life Experiences per each semester they are enrolled. These include a variety of cultural and intellectual events and must be sponsored by a student organization or department with attendance tracked on HeelLife. 
****

=== Travel Arrangements with Speakers ===
* A week or two prior, confirm travel plans with speaker (flights, hotels), doublecheck if they need pickup from the airport or hotel, or if parking needs to be arranged on/near campus

=== Equipment Needs ===
* A few days before speaker arrives, confirm equipment needs:
** Video camera/broadcasting equipment (i.e., for live-casting an in-person event on zoom for hybrid events)
** Lapel mike
** Laptop
** Presentation clicker/pointer
** Any adapters

=== Food and Room Reservations ===
* Food ordering placeholder
* Room Reservation placeholder

== Day-Of Planning ==
Arrive early for setup. Double check food orders and room reservations, and ensure all materials are printed/reviewed and available prior to meeting.

=== Technology Test ===
Do a technology test run to troubleshoot issues in advance
* The speaker should be there (if possible)
* Test sound and connectivity 
* Ensure all equipment is working properly and there are no additional adapters or accommodations needed. 
* If the talk is over zoom, schedule a test with the speaker in the location they will be in during the event.

=== Recording ===
* The best results would be to have the host of the zoom meeting recorded in the cloud.  Before doing this, make sure the following settings are enabled. 
** Under the settings page in the browser (not the app) Got to “Recording” Tab
** Under Cloud Recording, make sure that following settings are checked:
*** [x] Record active speaker with shared screen
*** [x] Record active speaker gallery view and shared screen separately 
**** [x] Active speaker
**** [x] Gallery View
**** [x] Shared Screen 
*** [x] Save chat messages from the meeting/ webinar
** Under Advanced cloud recording settings:
*** [  ] Add a timestamp to the recording (unchecked)
*** [x] Audio transcript 
** [test first] Allow cloud recording sharing 
* Upload video file MP4 to the appropriate speaker folder in the Google Drive

== After Presentation ==

=== Financial Documents ===
* Double check that all financial documents have been collected and follow-up as needed
* Ensure payment to each speaker in the amount listed on the Independent Contractor Services Agreement, through Zelle and the email address the speaker provides

=== Presentation Recordings ===
* Upload videos to YouTube folder on the Drive for editing and uploading
* Post MP4 files to HGAPS YouTube page
** Send private links to board, Eric for review
** After a week or all parties have reviewed, switch link to public and send courtesy link to all speakers who appear in the video

= Continuing Education Credit Additional Planning =
Like with standard speaker presentations, Continuing Education (CE) credit events should be planned well in advance with funding from the relevant Student Government Association or body from the host school. However, these events require a professional co-sponsoring organization accredited to provide CE certificates for event attendance. UNC-Chapel Hill has historically used the [https://www.ncpsychology.org/ North Carolina Psychological Association] (NCPA), but other organizations may serve as well. These organizations will have their own rules about event attendance, and may require information from HGAPS, the event speaker, and event attendees. CE Speakers are more expensive than regular speakers since co-sponsoring organizations will typically charge a fee per certificate issued. 

== Pre-Semester Planning ==

=== NCPA Co-Sponsorship Application Form ===
* Refer to the [https://www.ncpsychology.org/assets/docs/CE%20Co-Sponsor%20Manual%20March%202022%20%20WITH%20Appendices.pdf NCPA CE Manual] for guidance on filling out application
* Will require:
** Speaker bio
** Speaker contact info
** Lecture title and abstract
** 3-4 references
** 3-4 Learning Objectives (LOs): 3-4 LOs for a 1-3 hour program; 5-6 LOs for a 4-7 hour program
* Should be submitted 1 month or greater prior to date of presentation where possible
* Submit finalized form to Eric Youngstorm to sign, send signed form to Karen Gray (Director of Membership and Continuing Education): karen@ncpsychology.org at NCPA, cc’ing Eric Youngstrom

=== Additional Materials ===
NCPA requires verifications of attendance and engagement in order to award CE credit. To assist with processing attendance, the following forms should be prepared prior to the presentation, and linked in the presentation form itself (either as a slide in the speaker's PowerPoint or as an endplate slide): 
* Sign-in, Sign-out Sheets 
* [https://drive.google.com/file/d/1Wrx_uZIlEf1QSOnyPvJtyU2RFaJzvS_B/view?usp=sharing Qualtrics survey] to collect feedback about fulfillment of learning objectives and other comments

NCPA requires attendees to have signed in, signed out, and completed the survey to receive for CE credit, though exceptions are made if attendees have completed two of the three. 

== Week Prior to Presentation ==
* Get copy of speaker slides from speaker, if available
* Have bookend slides for PowerPoint presentations with QR code for Sign-In Sheet, sign-out sheet, and Qualtrics survey (at least by 24 hours prior)

== Day-Of Extra Steps ==
* Whoever introduces speaker should verbally explain this at beginning and end of presentation
* Have an extra person who can be co-host in case the host loses signal to ensure the recording continues
* Have someone available on zoom to moderate questions 

== After Presentation ==
* Send reminder emails to attendees to complete Qualtrics survey a few days after the presentation. All surveys should be complete by one week after the presentation date. 
* Get a final count on the amount of attendees requesting CE credit and send it to Dr. Youngstrom to process payment to NCPA for certificates
* Merge sign-in, sign-out, and Qualtrics survey form into one document and send to Karen Gray, cc’ing Dr. Eric Youngstrom. Allow 2 weeks for NCPA to complete surveys

=== CE Certificates ===
* NCPA will send personalized certificates to Dr. Youngstrom and the designated point of contact
* Certificates should be sent to attendees individually using an HGAPS email account

{{quotation| Sample email:
&lt;blockquote&gt;
Dear Colleague,

Attached please find your CE certificate for attending [SPEAKER NAME]’s seminar and completing the attendance and follow-up survey. Thank you for your patience as we worked with NCPA to provide the CE certificates. We also appreciate the help from Karen Gray at NCPA in facilitating the logistics.

We are already beginning plans for next semester’s speaker series, and we look forward to implementing some of your suggestions for topics then. 

Thanks for all you do for your clients and families, and thanks for your continued interest in HGAPS!

Best regards,

Eric Youngstrom &amp; the HGAPS Team
&lt;/blockquote&gt;}}</text>
      <sha1>he89cezvwetqqucf13n9wycxg7ojo7n</sha1>
    </revision>
  </page>
  <page>
    <title>User:Dc.samizdat/Real Euclidean four-dimensional space</title>
    <ns>2</ns>
    <id>289273</id>
    <revision>
      <id>2834705</id>
      <parentid>2834643</parentid>
      <timestamp>2026-09-27T20:06:46Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* An object's motion in space is the product of its discrete self-reflections */</comment>
      <origin>2834705</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101665" sha1="0np7ufn9eaz9n409h4bsmv8sbvk4h9j" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>0np7ufn9eaz9n409h4bsmv8sbvk4h9j</sha1>
    </revision>
    <revision>
      <id>2834711</id>
      <parentid>2834705</parentid>
      <timestamp>2026-09-27T20:35:04Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
      <origin>2834711</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101680" sha1="02qk3nka9c9wewlyserj95lanbglkq2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>02qk3nka9c9wewlyserj95lanbglkq2</sha1>
    </revision>
    <revision>
      <id>2834715</id>
      <parentid>2834711</parentid>
      <timestamp>2026-09-27T20:53:04Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
      <origin>2834715</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101734" sha1="lr22o3h9jzongg8pahyi54b1x0efgll" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>lr22o3h9jzongg8pahyi54b1x0efgll</sha1>
    </revision>
    <revision>
      <id>2834717</id>
      <parentid>2834715</parentid>
      <timestamp>2026-09-27T20:56:19Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
      <origin>2834717</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101747" sha1="ovdnq5ubafq0pxq93b210hbnjep505z" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ovdnq5ubafq0pxq93b210hbnjep505z</sha1>
    </revision>
    <revision>
      <id>2834718</id>
      <parentid>2834717</parentid>
      <timestamp>2026-09-27T21:02:42Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Symmetries */</comment>
      <origin>2834718</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101752" sha1="koibedrci7qcjyavjsj6rjmmsv7qzsq" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>koibedrci7qcjyavjsj6rjmmsv7qzsq</sha1>
    </revision>
    <revision>
      <id>2834720</id>
      <parentid>2834718</parentid>
      <timestamp>2026-09-27T21:13:49Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity describes Euclidean 4-space */</comment>
      <origin>2834720</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101768" sha1="gqi2z3ul04s5qo0dgbx0zpswpe6fip2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>gqi2z3ul04s5qo0dgbx0zpswpe6fip2</sha1>
    </revision>
    <revision>
      <id>2834721</id>
      <parentid>2834720</parentid>
      <timestamp>2026-09-27T21:19:12Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity describes Euclidean 4-space */</comment>
      <origin>2834721</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101770" sha1="dsle3gix705dyk0g5l7m080zb5mmap2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>dsle3gix705dyk0g5l7m080zb5mmap2</sha1>
    </revision>
    <revision>
      <id>2834723</id>
      <parentid>2834721</parentid>
      <timestamp>2026-09-27T21:31:39Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity describes Euclidean 4-space */</comment>
      <origin>2834723</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101812" sha1="2zaa5tbcf2m3myhhkysaewawiv7muot" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>2zaa5tbcf2m3myhhkysaewawiv7muot</sha1>
    </revision>
    <revision>
      <id>2834725</id>
      <parentid>2834723</parentid>
      <timestamp>2026-09-27T21:39:44Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity describes Euclidean 4-space */</comment>
      <origin>2834725</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101822" sha1="075qo0qvkn97r8icsaxbynlgzazbwyq" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw displacement, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>075qo0qvkn97r8icsaxbynlgzazbwyq</sha1>
    </revision>
    <revision>
      <id>2834727</id>
      <parentid>2834725</parentid>
      <timestamp>2026-09-27T22:14:00Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* An object's motion in space is the product of its discrete self-reflections */</comment>
      <origin>2834727</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101821" sha1="8ghinymnd5rqs0xlzpk0gazinfbywtk" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

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

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

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

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

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

== Symmetries ==

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

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

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

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

== Special relativity describes Euclidean 4-space ==

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

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

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

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

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

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

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

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

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

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

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

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

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame in relative motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

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

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

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

== Distribution of stars in our galaxy ==

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

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

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

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

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

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

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

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

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

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

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

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

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

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

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

== Revolutions ==

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

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

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

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

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

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

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

== Origins of the theory ==

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

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

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

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

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

== Boundaries ==

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

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Efn|name=radially equilateral}}

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

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

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

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

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

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

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

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

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

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

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

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>8ghinymnd5rqs0xlzpk0gazinfbywtk</sha1>
    </revision>
    <revision>
      <id>2834735</id>
      <parentid>2834727</parentid>
      <timestamp>2026-09-27T23:13:47Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* An object's motion in space is the product of its discrete self-reflections */</comment>
      <origin>2834735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101860" sha1="5l0lqevt9z64ihz2141azebc89sjn2b" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>5l0lqevt9z64ihz2141azebc89sjn2b</sha1>
    </revision>
    <revision>
      <id>2834764</id>
      <parentid>2834735</parentid>
      <timestamp>2026-09-28T01:29:55Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* An object's motion in space is the product of its discrete self-reflections */</comment>
      <origin>2834764</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101859" sha1="myycz44349y969ev1pkw1d8jzxneak4" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>myycz44349y969ev1pkw1d8jzxneak4</sha1>
    </revision>
    <revision>
      <id>2834765</id>
      <parentid>2834764</parentid>
      <timestamp>2026-09-28T01:34:24Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <origin>2834765</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101856" sha1="bz43bywmkbqygh66p1e3uxe7oqitdbq" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>bz43bywmkbqygh66p1e3uxe7oqitdbq</sha1>
    </revision>
    <revision>
      <id>2834766</id>
      <parentid>2834765</parentid>
      <timestamp>2026-09-28T01:36:22Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions */</comment>
      <origin>2834766</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101872" sha1="i2yluiayjqnj3tuydxgxg8h0el32jab" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>i2yluiayjqnj3tuydxgxg8h0el32jab</sha1>
    </revision>
    <revision>
      <id>2834769</id>
      <parentid>2834766</parentid>
      <timestamp>2026-09-28T01:40:41Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Revolutions */</comment>
      <origin>2834769</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101863" sha1="l46gcnkj13ly8yd4h4jltpebh1ajzrg" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>l46gcnkj13ly8yd4h4jltpebh1ajzrg</sha1>
    </revision>
    <revision>
      <id>2834771</id>
      <parentid>2834769</parentid>
      <timestamp>2026-09-28T01:43:29Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Revolutions */</comment>
      <origin>2834771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101866" sha1="4gq141chklwdi86dx04riawdg1q3eur" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>4gq141chklwdi86dx04riawdg1q3eur</sha1>
    </revision>
    <revision>
      <id>2834775</id>
      <parentid>2834771</parentid>
      <timestamp>2026-09-28T01:55:50Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Revolutions */</comment>
      <origin>2834775</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101918" sha1="gspigf40lwo4oh8w53wq7hlve4wrusr" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>gspigf40lwo4oh8w53wq7hlve4wrusr</sha1>
    </revision>
    <revision>
      <id>2834778</id>
      <parentid>2834775</parentid>
      <timestamp>2026-09-28T02:03:26Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Origins of the theory */</comment>
      <origin>2834778</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101921" sha1="j8exeeo6sgbsf529fvdti6rvtes06h2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>j8exeeo6sgbsf529fvdti6rvtes06h2</sha1>
    </revision>
    <revision>
      <id>2834789</id>
      <parentid>2834778</parentid>
      <timestamp>2026-09-28T04:11:11Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Origins of the theory */</comment>
      <origin>2834789</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101907" sha1="taddolbb4v4pgp81i4c625wvwtnpyxl" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;\mathbb{S}^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>taddolbb4v4pgp81i4c625wvwtnpyxl</sha1>
    </revision>
    <revision>
      <id>2834790</id>
      <parentid>2834789</parentid>
      <timestamp>2026-09-28T04:19:25Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Origins of the theory */</comment>
      <origin>2834790</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101923" sha1="kq27em30pjfekry6wzs0tj3yh1q6zhj" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space must be the actual geometry of our physical universe, and Minkowski spacetime an abstraction of it; our reciprocal of Einstein's disclaimer is the more direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>kq27em30pjfekry6wzs0tj3yh1q6zhj</sha1>
    </revision>
    <revision>
      <id>2834791</id>
      <parentid>2834790</parentid>
      <timestamp>2026-09-28T04:26:48Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Origins of the theory */</comment>
      <origin>2834791</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101940" sha1="mwyfmkqqak9bco13bo6zs422lgldach" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). The fact that the planets orbit on ellipses in our 3-space suggests that Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>mwyfmkqqak9bco13bo6zs422lgldach</sha1>
    </revision>
    <revision>
      <id>2834792</id>
      <parentid>2834791</parentid>
      <timestamp>2026-09-28T04:34:17Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Origins of the theory */</comment>
      <origin>2834792</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101982" sha1="f63xgn13hzrm48x2hpsrggg4bylfsxi" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>f63xgn13hzrm48x2hpsrggg4bylfsxi</sha1>
    </revision>
    <revision>
      <id>2834797</id>
      <parentid>2834792</parentid>
      <timestamp>2026-09-28T04:57:02Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions */</comment>
      <origin>2834797</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101987" sha1="9vkhwcozx4jb74zhlcwbh19ql3dgw6p" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>9vkhwcozx4jb74zhlcwbh19ql3dgw6p</sha1>
    </revision>
    <revision>
      <id>2834799</id>
      <parentid>2834797</parentid>
      <timestamp>2026-09-28T04:57:38Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions */</comment>
      <origin>2834799</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101969" sha1="o696duamr8zlwywdqey2jxf6f7l3o7r" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>o696duamr8zlwywdqey2jxf6f7l3o7r</sha1>
    </revision>
    <revision>
      <id>2834801</id>
      <parentid>2834799</parentid>
      <timestamp>2026-09-28T05:04:26Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Distribution of stars in our galaxy */</comment>
      <origin>2834801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="101972" sha1="lspzm7z49veueazddxq6nxaeoynnrt8" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - September 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space has four perpendicular dimensions, not just three.
* Atoms are [[W:4-polytope|4-polytopes]].
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions.
* Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space.
* Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.
* Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space at velocity &lt;math&gt;c&lt;/math&gt; on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation &lt;math&gt;\mathrm{T}&lt;/math&gt; rather than a rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. 

Four dimensional Euclidean space can contain no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion.

In a universe which is four-dimensional Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, wherein these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion....{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ==

{{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}}

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery.

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;(n+1)&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>lspzm7z49veueazddxq6nxaeoynnrt8</sha1>
    </revision>
  </page>
  <page>
    <title>Category:Hatnote templates with errors</title>
    <ns>14</ns>
    <id>290980</id>
    <revision>
      <id>2834812</id>
      <parentid>2456770</parentid>
      <timestamp>2026-09-28T06:30:04Z</timestamp>
      <contributor>
        <username>Samsulkj9</username>
        <id>3111373</id>
      </contributor>
      <origin>2834812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="71" sha1="304p61bax8o2qm5jdbc5n7yth7y00qm" xml:space="preserve">{{Empty category}}
{{center|{{purge|UPDATE this page}}}}
__showdetail__</text>
      <sha1>304p61bax8o2qm5jdbc5n7yth7y00qm</sha1>
    </revision>
    <revision>
      <id>2834824</id>
      <parentid>2834812</parentid>
      <timestamp>2026-09-28T09:07:29Z</timestamp>
      <contributor>
        <username>NguoiDungKhongDinhDanh</username>
        <id>2922842</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/Samsulkj9|Samsulkj9]] ([[User_talk:Samsulkj9|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]]</comment>
      <origin>2456770</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="70" sha1="suv5ipg1ysuk65so3i6n700p9bak79w" xml:space="preserve">{{Empty category}}
{{center|{{purge|UPDATE this page}}}}
__HIDDENCAT__</text>
      <sha1>suv5ipg1ysuk65so3i6n700p9bak79w</sha1>
    </revision>
  </page>
  <page>
    <title>Global Audiology/Asia/Hong Kong</title>
    <ns>0</ns>
    <id>296049</id>
    <revision>
      <id>2834681</id>
      <parentid>2806762</parentid>
      <timestamp>2026-09-27T15:17:39Z</timestamp>
      <contributor>
        <username>PhantomFunds</username>
        <id>3111705</id>
      </contributor>
      <origin>2834681</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12708" sha1="sgpl6roz2h66bgs0azpsklycn6edtil" xml:space="preserve">{{:Global Audiology/Header}}{{:Global Audiology/Asia/Header}}{{CountryHeader|File:Hong Kong on the globe (Hong Kong centered with inset).svg|https://en.wikipedia.org/wiki/Hong_Kong}}{{HTitle|Brief Country Information }} 
[https://en.wikipedia.org/wiki/Hong_Kong Hong Kong] is a special administrative region of China. Situated on China's southern coast just south of Shenzhen, it consists of Hong Kong Island, Kowloon, and the New Territories. Chinese and English are the official languages of Hong Kong. English is widely used in the Government and by the legal, professional, and business sectors. Trilingual professionals who speak English, Cantonese, and Putonghua (Mandarin Chinese) play a vital role in the numerous professional fields in Hong Kong. 

{{HTitle|Incidence and Prevalence of Hearing Loss }}A territory-wide survey on persons with disabilities and chronic diseases was conducted during August 2019 - December 2020 via the local continuous sample survey, the General Household Survey, to estimate the total number and prevalence rate of persons with selected types of disabilities and chronic diseases. 

In 2020, some 266,900 people (or 3.6% of the total population) in Hong Kong reported that they had some difficulty in hearing, 44,300 people (0.6%) had a lot of difficulty, and 3,600 people (less than 0.05%) could not hear at all. Among the total population, some 53,400 people (0.7%) reported using a hearing aid or tool.

Among the 266,900 people who reported that they had some difficulty in hearing, 9.4% cited that they were using a hearing aid or cochlear implant. Among the 44,300 people who had a lot of difficulty in hearing, 24.0% cited that they were using a hearing aid or cochlear implant. And among the 3,600 people who could not hear at all, 9.3% cited that they were using a hearing aid or cochlear implant. Of the 47,900 people aged 2 and over with hearing difficulty, 3,000 (6.3%) reported that they use sign language in their usual communication.

In Hong Kong, disability statistics are mainly available from relevant Government bureaus or departments, statutory bodies, and nongovernmental organizations. In particular, the Central Registry for Rehabilitation of the Labour and Welfare Bureau issues the Registration Card for People with Disabilities (Registration Card) to persons with permanent or temporary disabilities as a documentary proof of their disability status and maintains relevant records. Persons with disabilities and with certification by registered medical practitioners or allied health professionals, etc., could apply for the Registration Card. There were some 93,000 holders of valid Registration cards as of March 2021. As such applications are on a voluntary basis, it is assumed that such figures are some sort of lower bound estimates of the number of people with the specific type of disability in Hong Kong.²

{{HTitle|Information About Audiology}}
=== Educational Institution ===
The Faculty of Education at the University of Hong Kong was one of the first in Southeast Asia and China to offer post-graduate training in audiology. The first students were enrolled in 1996, with intakes every two years. The two-year program provides students with detailed theoretical background knowledge of human hearing and hearing loss, as well as intensive clinical practice in a variety of audiology placements. The medium of instruction is English. The program is designed to provide students with high-quality clinical skills and an appreciation of research in audiology.³

=== Professional Bodies ===

==== Hong Kong Society of Audiology ====

The Hong Kong Society of Audiology is a voluntary organization founded in 1992 by a group of audiologists in Hong Kong. The Society has grown steadily over the years. In 2023, The Society has approximately 120 professional members.⁴

The Hong Kong Society of Audiology Limited endeavours:
* To share among members the information and technology in Audiology and other related fields so as to update and promote the standard of audiological assessments and rehabilitation services in Hong Kong.
* To promote research in the area of Audiology and related fields with reference * To local needs for study, diagnosis, alleviation, and prevention of hearing impairment.
* To serve as a channel of communication among members and local and overseas professional bodies in matters related to Audiology, education, hearing, and speech sciences.
* To serve as a consultative body for other professional organisations and community bodies on matters related to Audiology education, hearing, and speech sciences.
* To serve as a social group for members of the Society and to promote their welfare in works related to Audiology, education, hearing, and speech sciences.

==== Hong Kong Institute of Audiologists ====

The Hong Kong Institute of Audiologists (HKIA) was established in March 2018 to enhance the professional standard of audiologists in Hong Kong and to participate in the Accredited Registers Scheme for Healthcare Professions administered by the Government of the Hong Kong Special Administrative Region (6). HKIA ensures that the local public has access to professional audiological assessment and rehabilitation services provided by qualified audiologists.  The Institute has approximately 60 audiologist registrants in 2023.⁵

The missions of the HKIA include:
* To maintain professional standards of audiology services in Hong Kong;
* To safeguard the public's interests in accessing audiology services in Hong Kong;
* To maintain adequate standards of professional practice for audiologists;
* To promote adequate standards of professional practice and of professional conduct among audiologist registrants;
* To establish and maintain the accredited register of audiologists in Hong Kong;
* To establish and maintain contact with other members of the audiology profession in Hong Kong.

{{HTitle|Research in Audiology}}The majority of audiology research in Hong Kong is led by research teams from The University of Hong Kong, Education University of Hong Kong, the Chinese University of Hong Kong. 

{{HTitle|Audiology practice}}
==== Elderly Health Care Voucher Scheme ====

The coverage of the Elderly Health Care Voucher Scheme (EHVS) in Hong Kong was extended to include primary healthcare services provided by four categories of the healthcare profession under the Accredited Registers Scheme for Healthcare Professions (i.e., audiologists, dietitians, clinical psychologists, and speech therapists), as well as medical equipment (such as hearing aids) provided by them upon professional assessment in April 2023.
 
It is estimated that almost 1.7 million eligible elderly people in Hong Kong benefit from the EHVS. Together with the four new categories of healthcare professions, eligible elderly persons are able to make use of vouchers to pay for primary healthcare services provided by a total of 14 categories of healthcare professions (i.e., medical practitioners, Chinese medicine practitioners, dentists, nurses, physiotherapists, occupational therapists, radiographers, medical laboratory technologists, chiropractors, optometrists with Part I registration, audiologists, dietitians, clinical psychologists, and speech therapists). As of March 2023, over 11,000 healthcare service providers in Hong Kong had already enrolled in the EHVS. Eligible elderly persons can use vouchers at over 33,000 service points across the territory.
 
Under the EHVS, eligible persons aged 65 and above are given an annual voucher amount to pay for services provided by private primary healthcare service providers enrolled in the EHVS. Any unused voucher amount can be carried forward for use in the following years, subject to a maximum accumulation limit.⁷
{{HTitle|Audiology Charities}}
==== The Hong Kong Society for the Deaf ====
Found in 1968, the Hong Kong Society for the Deaf is a non-profit making organisation which aims to promote the well-being of the hearing impaired and seeks to ensure equalisation of opportunities for hearing impaired persons. The Society aims to provide comprehensive and professional services of the highest standards to ensure equalisation of opportunities for the hearing impaired, and to promote self-development, self-actualisation and self-sufficiency among the hearing impaired so they could integrate into society. Its objectives include:
* To undertake projects of publicity, education, recreation, counseling, audiological and medical services for the hearing impaired; and to assist or collaborate with any institutions, organisations or individuals to improve the services for the hearing impaired.
* To work towards improving the educational standards for the hearing impaired, to provide scholarships and special equipment for the hearing impaired, and to provide guidance for parents of hearing impaired children.
* To inform the public about the problems and needs of the hearing impaired, to give necessary information to hearing impaired persons and their families about institutions and services available to them, and to exchange information among institutions serving the hearing impaired and to work towards the integration of the hearing impaired with the general public.⁸

==== Hear Talk Foundation ====
Launched in 2003, Hear Talk Foundation is a registered charitable organisation in Hong Kong committed to serving the underprivileged communities with hearing impairment and speech disorders both in Hong Kong and Mainland China, especially children and the elderly. It has been established by a group of committed ENT specialists, audiologists, speech therapists, and educators.⁹

{{HTitle|Challenges, Opportunities and Notes}}
=== Challenges ===
* There is a shortage of manpower in audiology, both in clinical practices and in the academic field.
* Health literacy and cultural factors are the main barriers to the utilization of hearing health services.
* The majority of the local population speaks Cantonese Chinese as their native language. Only a very limited number of speech assessment materials (including speech recognition tests, speech perception tests, and speech audiometry) are available in the local language.
* There is no regulation for any hearing aid or hearing device.  There is also no statutory registration for anyone practicing in the audiology field.

{{HTitle|References}}
{{reflist}}

[[Category:Audiology]]
[[Category:Hong Kong]]
&lt;ref&gt;{{Cite web|url=https://www.gov.hk/en/about/abouthk/facts.htm|title=GovHK: Hong Kong – the Facts|last=GovHK (www.gov.hk)|website=www.gov.hk|language=en|access-date=2023-06-26}}&lt;/ref&gt;&lt;ref&gt;Census and Statistics Department, Hong Kong SAR. Social data collected via the General Household Survey Special Topics Report No. 63: Persons with disabilities and chronic diseases. 2021; Available at: &lt;nowiki&gt;https://www.censtatd.gov.hk/en/data/stat_report/product/C0000055/att/B11301632021XXXXB0100.pdf&lt;/nowiki&gt;. Accessed Jun 21, 2023.&lt;/ref&gt;&lt;ref&gt;Faculty of Education, The University of Hong Kong. Master of Science in Audiology. 2023; Available at: &lt;nowiki&gt;https://web.edu.hku.hk/programme/audiology&lt;/nowiki&gt;. Accessed Jun 21, 2023.&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.audiology.org.hk/about/our-mission/|title=HKSA and Our Mission – Hong Kong Society of Audiology|language=en-GB|access-date=2023-06-26}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.audiologists.org.hk/about-hkia-2/|title=Audiologists 聽力學家 – Hong Kong Institute of Audiologists|language=en-GB|access-date=2023-06-26}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.ars.gov.hk/en/accr_pro_bodies.html|title=Accredited Registers Scheme for Healthcare Professions - Accredited Healthcare Professional Bodies|website=www.ars.gov.hk|access-date=2023-06-26}}&lt;/ref&gt;&lt;ref&gt;Press Releases, The Government of the Hong Kong SAR. Coverage of Elderly Health Care Voucher Scheme to extend to include four categories of healthcare profession under Accredited Registers Scheme for Healthcare Professions. 2023; Available at: &lt;nowiki&gt;https://www.info.gov.hk/gia/general/202304/27/P2023042700410.htm&lt;/nowiki&gt;. Accessed Jun 21, 2023.&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.deaf.org.hk/en/mission.php|title=The Hong Kong Society for the Deaf|website=www.deaf.org.hk|access-date=2023-06-26}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://www.heartalk.org/en/about_us/our_mission/|title=Our Mission {{!}} Hear Talk Foundation|website=www.heartalk.org|access-date=2023-06-26}}&lt;/ref&gt;
&lt;references /&gt;

{{Global Audiology Authors
|name1=NG Hoi Yee Iris
|role1=Author
|website1=http://www.ihcr.cuhk.edu.hk/professor-iris-hoi-yee-ng/
|name2=Joyce Rodvie Sagun
|role2=Contributor
|linkedin2=http://linkedin.com/in/joyce-rodvie-sagun-4691bb182
}}</text>
      <sha1>sgpl6roz2h66bgs0azpsklycn6edtil</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Youth environmental activism motivation</title>
    <ns>0</ns>
    <id>307211</id>
    <revision>
      <id>2834817</id>
      <parentid>2834668</parentid>
      <timestamp>2026-09-28T06:56:02Z</timestamp>
      <contributor>
        <username>~2026-49676-17</username>
        <id>3111017</id>
      </contributor>
      <comment>/* Overview */</comment>
      <origin>2834817</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20994" sha1="ksx25l1c5gsbd4niwk4jlqb9t6mem2a" xml:space="preserve">{{title|Youth environmental activism motivation:&lt;br&gt;What motivates young people to engage in environmental activism?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
'''Case Study 1: 2010 Study on Chile Environmental Activism'''

Recently there have been growing concerns about mining and its risks for individuals as well as the environment. The mining of gold, has a high environmental impact. This impact has increased over time as new and riskier technologies have been developed (Mudd, 2007). One of the biggest environmental impacts of open pit mining is high water consumption and acid mine drainage (Urkidi, 2010) . Socially studies have shown that mining areas tend to have comparatively higher levels of economic inequity, AIDS, alcoholism, prostitution and child labour, accentuating poverty and social conflict (Pegg, 2006). 

In Chile from 2009 to 2019, Scherman and colleagues (2022) wrote about a fluctuation of environmental activism in Chili due to the above conditions. During this period they saw a drastic rise in a specific kind of people. Younger generations predominated among environmental activists, who were particularly active (Scherman et al., 2022). This correlated with social media and different ways to protest through instant messaging and sharing ideas. Scherman and colleagues (2022) found social media use was positively associated with participation in environmental issues. Because of the large scale of protesters, they were able to block mining and energy work from being continued. What other motivators do youth have for environmental activism? More engaging?{{grammar}}{{RoundBoxBottom}}

(Figure 1.) Chilian People protesting against a mine in Chile.

insert image below

'''Problem statement''': Although Climate change awareness is increasing in all people including Youth Climate change is still continuing rapidly. How can individuals especially youth help combat this?

'''Key points'''
* The use of social media as a motivation in environmental issues.
* Rise of youth in protesting.
*Reasons people protest; Social unrest, environmental risks and individual risks.
*Key definitions
*
'''Focus questions:''' Establish [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

{{RoundBoxTop|theme=3}} '''Focus questions'''

*Why are more youth participating in activism? same question as below have another
*What is motivating youth to engage in activism?
*What role does nature, nurture and culture play on youth activism?
*What is the relationship between education and environmental activism?
*How can we motivate young people to engage in environmental activism?
{{RoundBoxBottom}}

What is [https://www.liberties.eu/en/stories/activism/44871 activism]?

What is [https://www.liberties.eu/en/stories/activism/44871 environmental activism]?

What is [[motivation]]?

== History of motivation and activism ==
Activism can be traced back to the Enlightenment era during the 17th and 18th centuries (Global Society World News., 2024) . From there the first form of environmental activism can be traced back to ancient civilizations, where practices like soil conservation were established as early as 2,000 years ago in China, India, and Peru. However, the modern environmental movement began to take shape in the late 19th century, primarily in response to pollution and industrialization, leading to significant legislative changes and public awareness. Notable figures like John Muir and Aldo Leopold played crucial roles in advocating for the preservation of natural landscapes and the conservation of resources, influencing the development of the contemporary environmental movement. (Elliot, Britannica., 2026)

Activist movements have played a significant role in shaping history and pushing for social and political change (Luttrell., 2023)

Motivation can be influenced by others and the environment. Individuals are motivated to participate in activism due to poor conditions, for social justice work, but also for sanctuary and peer/adult relationships. (Akiva et al., 2017)

The prevalence of environmental activism has increased in the past century due to more in depth knowledge about our environment and climate change. 

;Key points
* Activism and environmental activism has been happening for centuries 
* Young people have not always been interested in activism 
* There have always been causes to act for or against 
* Motivations are often for a better world

== The landscape ==
What are the contributing factors for younger generations to be so active on environmental activism? There is a rising trend of youth groups being formed with the explicit intent of educating other peers and elders on the social and political implications of climate change (Goldman et al., 2014; Feldmann, 2020). 

What motivates this generation to actively protest? 

Today's landscape that encourages youth environmental action is one of publicized climate action, social change and.....  

=== 2010s ===
Climate activism has been around for much longer than 2010-2019. However, there was a clear shift in the amount of people listening these years, and the way it was portrayed in the media. Throughout these years it was clear that it climate change was happening. It wasn't a concern, it wasn't something oncoming, it was here. There was an emergence of not only regular climate protests {{clarify}} (See Figure 1), but they began to emerge on a global scale (Scherman et al., 2022). Rather than them staying entirely within individual groups, social media forced it to be within the public conscious.... More examples of how media and public awareness shifted during the 2010s.

[[File:3rd Global Climate Strike Berlin FridaysForFuture demonstration view from stage 50.jpg|thumb|(Figure 1.) Climate strike ]]

=== Social media ===
* Clear rise in global awareness due to the broad scope of [[wikipedia:Social_media|social media]]. (Scherman et al., 2022).
* Information is quickly passed around in comparison to previous generations.
* Able to see the effects at a rapid pace, rather than articles and TV programs.
* They are constantly aware of how it will effect their future.
* Expand on how social media plays a crucial role in youth activism (Platforms and campaigns)

=== Politics ===
* A more politically active generation (Scherman et al., 2022).
* Awareness of how these policies effect the environment. (Sloam et al., 2022).
What exactly makes the youth care so much about politics? Well, there has been a clear shift in the way they view politics, it is no longer something to worry about during elections, it is expected to have some form of political awareness (Sloam et al.,  2022).

* Role of politics needs more exploration....
* How are young activists influence policy and what political movements they are engaging with?

;Key Points

* Youth groups formed to encourage activism
* Landscape that encourages environmental action
* Shift in activism due to social media 
* A more political active generation 

== Motivation ==
So what are the things that actually motivate youths into such activism? 

* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

Well throughout the literature there are clear indicators of:

* Passion
* Awareness
* Goals
* [[Motivation and emotion/Book/2011/Self-efficacy|Self efficacy]]
The current youth have a clear understanding of environmental issues, and have a clear level of self efficacy towards them (Goldman, D. et al). They believe that the engaging with social media, having the conversations and living sustainable lives will make a difference....

There is a current trend of the younger generation being politically inexperienced, when they protest against specific decisions and have strikes which brings an emotional aspect into the fold (Fedmanm. H. 2020). They are angry....{{expand}} (Add more context about the emotional aspect of youth activism.)

Gousse-Lessard (2013) proposed that passion is a strong motivator behind radical environmental activism.

Being told they don't know what they're talking about makes personal stakes, mixed with a genuine care for the future. {{rewrite}} (Clarify the role of criticism and how it motivates further activism.)

Need more of a detailed analysis (Expanding on each motivator)

Emotional aspect of youth activism. 

This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise. Gradually remove these suggestions as the chapter develops. It is OK to retain some of this template content for the topic development exercise. Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].

The Overview is typically consists of one to four paragraphs inbetween the scenario and focus questions. Suggested word count aim for the Overview: 180 to 330 words.

;Key points ==

* Motivators of passion, awareness, goals, and self efficacy
* Passion and its role in radical environmental activism
* Youth trying to prove themselves 

==Role of future security as a motivator==
* [[File:Future_sustainabilty.png|thumb|Figure 2. Future of environmental sustainability]]Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but
** avoid having sections with only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
*Activism is motivated by a desire to make contributions to a just world (Wetering &amp; Lee., 2025) 
*Climate change is no longer something that can be ignored (Romano et al., 2024) 

*Young people will experience stronger consequences of climate change in their future adult lives than older people are facing today (Wallis and Loy., 2021). 
*Do it ourselves politics (Pickard., 2022) 

;Key points
* Youth fighting for their future 
*More obvious climate and environmental changes
*Knowledge of the impacts of climate change for the youth 

==Does culture motivate youth activism? ==
* Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but
** avoid having sections with only one sub-heading
** provide an introductory paragraph before breaking into sub-sections

;Key points
* Culture impacting young peoples beliefs 
* How people were raised impacting their activism 
*Different cultures valuing volunteering and activism more

==How does education impact activism? ==
* Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but
** avoid having sections with only one sub-heading
** provide an introductory paragraph before breaking into sub-sections

;Key points
* Higher educated more compelled to act 
* Knowledge of successful activism 
* Knowledge of future impacts if activism is not successful 

=== Integration (10%) ===

* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies

*

'''Research (25%)'''

* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]])
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* Critical analyse the key research findings, including limitations and implications

*

==Learning feature==
;Quiz (Add 7 more harder questions)
&lt;quiz display=&quot;simple&quot;&gt;
{Why were there protests in Chili:
|type=&quot;()&quot;}
+ Risks associated with mining 
- Climate change 
-Bushfires 
-Political corruption

{The protests in Chili got young people involved due to social media:
|type=&quot;()&quot;}
+True
- False

{What motivates young people to participate in environmental activism:
|type=&quot;()&quot;}
- Social media 
- Climate change 
- Passion 
+ All of the above 

&lt;/quiz&gt;

=== Learning features (5%) ===

* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** emotion involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &lt;nowiki&gt;&amp;lt;nowiki&amp;gt;&lt;/nowiki&gt;[[Motivation and emotion/Assessment/Chapter#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter

==Conclusion==
* Suggested word count: 150 to 330 words
* It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a pretty good idea of the problem and what is known based on psychological science
In conclusion Youth engaging in environmental activism are motivated by....

=== Conclusion (5%) ===

* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]
{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages? (Even for the topic development, have a go at the likely take-home message)
}}

==See also==
* [[wikipedia:Climate_movement|Climate Movement]]
* [[Climate change|Climate Change]]
{{tip|Suggestions for this section:
* Present in alphabetical order
* Use [[w:Letter case#Sentence casing|sentence casing]]
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Akiva, T., Carey, R. L., Cross, A. B., Delale-O'Connor, L., &amp; Brown, M. R. (2017). Reasons youth engage in activism programs: Social justice or sanctuary? Journal of Applied Developmental Psychology, 53, 20-30. https://doi.org/https://doi.org/10.1016/j.appdev.2017.08.005 

Elliott, L. (2026, May 11). environmentalism. Encyclopedia Britannica. https://www.britannica.com/topic/environmentalism

Feldman, H. R.. (2020). A rhetorical perspective on youth environmental activism. ''Journal of Science Communication'', ''19''(06), C07. https://doi.org/10.22323/2.19060307

Goldman, D., Pe’er, S., &amp; Yavetz, B. (2015). Environmental literacy of youth movement members – is environmentalism a component of their social activism? ''Environmental Education Research'', ''23''(4), 486–514. https://doi.org/10.1080/13504622.2015.1108390

Gousse-Lessard, A.-S., Vallerand, R. J., Carbonneau, N., &amp; Lafrenière, M.-A. K. (2013). The role of passion in mainstream and radical behaviors: A look at environmental activism. ''Journal of Environmental Psychology'', ''35'', 18–29. https://doi.org/10.1016/j.jenvp.2013.03.003

Luttrell, R. (2023). Historical Roots and Modern Movements: A Framework for Activism. In Strategic Social Media as Activism (pp. 5-39). Routledge.

Pickard, S. (2022). Young environmental activists and Do-It-Ourselves (DIO) politics: Collective engagement, generational agency, efficacy, belonging and hope. Journal of Youth Studies, 25(6), 730-750.

Romano, L., Russo, C., Gladwin, T. E., &amp; Panno, A. (2024). Adolescents and young adults’ participation in pro-environmental movements: A systematic review. The Journal of Genetic Psychology, 185(5), 373-398. 

Scherman, A., Valenzuela, S., &amp; Rivera, S.. (2022). Youth environmental activism in the age of social media: the case of Chile (2009-2019). ''Journal of Youth Studies'', ''25''(6), 751–770. https://doi.org/10.1080/13676261.2021.2010691

Sloam, J., Pickard, S., &amp; Henn, M.. (2022). ‘Young People and Environmental Activism: The Transformation of Democratic Politics’. ''Journal of Youth Studies'', ''25''(6), 683–691. https://doi.org/10.1080/13676261.2022.2056678

Tagkaloglou, S., &amp; Kasser, T. (2018). Increasing collaborative, pro-environmental activism: The roles of Motivational Interviewing, self-determined motivation, and self-efficacy. ''Journal of Environmental Psychology'', ''58'', 86–92. https://doi.org/10.1016/j.jenvp.2018.06.004

Urkidi, L. (2010). A glocal environmental movement against gold mining: Pascua–Lama in Chile. Ecological Economics, 70(2), 219-227. https://doi.org/https://doi.org/10.1016/j.ecolecon.2010.05.004

Wallis, H., &amp; Loy, L. S. (2021). What drives pro-environmental activism of young people? A survey study on the Fridays For Future movement. Journal of Environmental Psychology, 74, 101581. https://doi.org/https://doi.org/10.1016/j.jenvp.2021.101581 

Wetering, J. V. D., &amp; Lee, K. (2026). What's in It for Them? A Developmental Science Perspective on Adolescent Climate Activism. Journal of Adolescence.

https://www.globalsociety.earth/post/the-origins-of-the-activism-movement-a-historical-and-global-perspective

}}

List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].APA style example

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Present in alphabetical order
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Activism]]
[[Category:Motivation and emotion/Book/Environment]]
[[Category:Motivation and emotion/Book/Youth]]</text>
      <sha1>ksx25l1c5gsbd4niwk4jlqb9t6mem2a</sha1>
    </revision>
  </page>
  <page>
    <title>User:Michael Ten/common.css</title>
    <ns>2</ns>
    <id>316385</id>
    <revision>
      <id>2834803</id>
      <parentid>2834214</parentid>
      <timestamp>2026-09-28T05:16:11Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <origin>2834803</origin>
      <model>css</model>
      <format>text/css</format>
      <text bytes="2195" sha1="bre2t9cmntlohsashcn3gstifb2efvs" xml:space="preserve">:root {
    --mt-text: #d4c4a8;
    --mt-link: #7eabd4;
    --mt-visited: #7aada0;
    --mt-hover: #b4d0e8;
    --mt-new: #e05656;
    --mt-new-hover: #ff8080;
}

.mw-body,
.mw-body-content,
.vector-body,
.mw-parser-output,
#firstHeading,
.firstHeading,
.mw-heading,
.mw-body h1,
.mw-body h2,
.mw-body h3,
.mw-body h4,
.mw-body h5,
.mw-body h6,
.vector-menu-heading,
.vector-pinnable-header-label,
.mw-footer,
#footer-info,
#footer-places {
    color: var(--mt-text) !important;
    -webkit-text-fill-color: var(--mt-text);
}

a:link {
    color: var(--mt-link) !important;
    -webkit-text-fill-color: var(--mt-link) !important;
}
a:visited {
    color: var(--mt-visited) !important;
    -webkit-text-fill-color: var(--mt-visited) !important;
}
a:hover {
    color: var(--mt-hover) !important;
    -webkit-text-fill-color: var(--mt-hover) !important;
}

a.new:link,
a.new:visited {
    color: var(--mt-new) !important;
    -webkit-text-fill-color: var(--mt-new) !important;
}
a.new:hover {
    color: var(--mt-new-hover) !important;
    -webkit-text-fill-color: var(--mt-new-hover) !important;
}

.mw-parser-output a.external:link,
.mw-parser-output a.external:visited {
    color: var(--mt-visited) !important;
    -webkit-text-fill-color: var(--mt-visited) !important;
}

#mw-panel .portal,
#mw-panel .body,
#mw-panel .vector-menu-content {
    margin: 0;
    padding: 0;
}
#mw-panel a,
#mw-panel .vector-menu-content a {
    margin: 2px 0;
    padding: 2px 5px;
    line-height: 1.2;
}

.mw-enhanced-rc,
.mw-enhanced-rc-nested {
    font-size: 1.15em;
}

@media screen and (min-width: 1000px) {
    .vector-feature-main-menu-pinned-enabled .mw-page-container-inner {
        grid-template-columns: 10rem minmax(0, 1fr);
    }
}

.mw-logo-icon,
.mw-logo-wordmark,
.mw-logo-tagline,
.mw-logo img {
    filter: invert(1) brightness(1.25);
}

.mw-enhancedchanges-arrow {
    filter: invert(1);
}

.wikiEditor-ui-top .oo-ui-iconElement-icon,
.wikiEditor-ui-top .oo-ui-indicatorElement-indicator,
.wikiEditor-ui-top .tool-button span,
.oo-ui-toolbar-tools .oo-ui-iconElement-icon,
.oo-ui-toolbar-tools .oo-ui-indicatorElement-indicator {
    background-color: transparent;
    filter: invert(1);
}</text>
      <sha1>bre2t9cmntlohsashcn3gstifb2efvs</sha1>
    </revision>
    <revision>
      <id>2834805</id>
      <parentid>2834803</parentid>
      <timestamp>2026-09-28T05:17:20Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <origin>2834805</origin>
      <model>css</model>
      <format>text/css</format>
      <text bytes="2181" sha1="n7qwnfynj0otpudjzzfqqkpsujbxd6f" xml:space="preserve">/*

 */

:root {
    --mt-link: #057cfc;
    --mt-visited: #02ddc4;
    --mt-hover: #86cafe;
    --mt-new: #ff2222;
    --mt-new-hover: #cc0000;
}

/* Links — :link avoids styling named anchors; !important still needed vs Dark Reader */
a:link {
    color: var(--mt-link) !important;
}
a:visited {
    color: var(--mt-visited) !important;
}
a:hover {
    color: var(--mt-hover) !important;
}

/* Uncreated pages */
a.new:link,
a.new:visited {
    color: var(--mt-new) !important;
}
a.new:hover {
    color: var(--mt-new-hover) !important;
}

/* External links: teal on the &lt;a&gt; only — do not touch descendants */
.mw-parser-output a.external:link,
.mw-parser-output a.external:visited {
    color: var(--mt-visited) !important;
    -webkit-text-fill-color: var(--mt-visited);
}

/* Compact left sidebar (legacy #mw-panel + Vector menus) */
#mw-panel .portal,
#mw-panel .body,
#mw-panel .vector-menu-content {
    margin: 0;
    padding: 0;
}

#mw-panel a,
#mw-panel .vector-menu-content a {
    margin: 2px 0;
    padding: 2px 5px;
    line-height: 1.2;
}

/* Watchlist / recent-changes density */
.mw-enhanced-rc,
.mw-enhanced-rc-nested {
    font-size: 1.15em;
}

/* Narrow pinned main menu — desktop only so mobile grid is untouched */
@media screen and (min-width: 1000px) {
    .vector-feature-main-menu-pinned-enabled .mw-page-container-inner {
        grid-template-columns: 10rem minmax(0, 1fr);
    }
}

/*
 * Icons / wordmark
 * One invert on a tiny image is cheap. Multi-step hue/sepia filters are not.
 */
.mw-logo-icon,
img[src=&quot;/static/images/mobile/copyright/wikiversity-wordmark-en.svg&quot;] {
    filter: invert(.65);
}

/* Enhanced RC expand/collapse arrows — few nodes, RC pages only */
.mw-enhancedchanges-arrow {
    filter: invert(1);
}

/*
 * Edit-toolbar glyphs (WikiEditor + OOUI).
 * invert(1) is one compositor op; the old 6-function chain was not.
 */
.wikiEditor-ui-top .oo-ui-iconElement-icon,
.wikiEditor-ui-top .oo-ui-indicatorElement-indicator,
.wikiEditor-ui-top .tool-button span,
.oo-ui-toolbar-tools .oo-ui-iconElement-icon,
.oo-ui-toolbar-tools .oo-ui-indicatorElement-indicator {
    background-color: transparent;
    filter: invert(1);
}</text>
      <sha1>n7qwnfynj0otpudjzzfqqkpsujbxd6f</sha1>
    </revision>
  </page>
  <page>
    <title>User:Jaspershields</title>
    <ns>2</ns>
    <id>322669</id>
    <revision>
      <id>2834695</id>
      <parentid>2766529</parentid>
      <timestamp>2026-09-27T18:46:52Z</timestamp>
      <contributor>
        <username>Jaspershields</username>
        <id>3005613</id>
      </contributor>
      <comment>/* About me */</comment>
      <origin>2834695</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5253" sha1="gx27gaaog61ds4hki03rtwlcl8kqn1x" xml:space="preserve">jasper shields was previously a full time student at the [https://www.canberra.edu.au/ University of Canberra], graduating with decent grades towards the end of 2025. he moved up to the gold coast to continue his education and study psychology honours.[[File:JasperShields.jpg|thumb|'''Figure 1.''' Jasper in 2024]]
he thinks this unit was very good, and thanks dr. neill for his effort and attention in preserving the work of students.

== book chapter ==
throughout the semester, he worked on his book chapter titled &quot;[[Motivation and emotion/Book/2025/Lighting and mood|Lighting and mood]]&quot;. 

it's worth mentioning that the current version of the chapter has been edited post marking to fix grammar and spelling mistakes, but is otherwise unchanged from submission.

== more? ==
he has a [https://www.youtube.com/channel/UChpV_jZSL9aI8ByS1ktuFiA YouTube channel,]containing some of his previous work from his time at the University of Canberra, and he is contactable on his [https://www.linkedin.com/in/jaspershields/ Linkedin]. in his free time, he plays chess, and makes levels for the free to play video game osu!.

== contributions ==
{| class=&quot;wikitable sortable mw-collapsible&quot;
!#
!Action
!Description
!Book chapter
!Date
|-
|1
|Comment
|[[Talk:Motivation and emotion/Book/2025/Neurodiversity and emotion#Autism Spectrum Disorder and Emotional Regulation|Made suggestion regarding Autism Spectrum Disorder (ASD) and Emotional Regulation]]
|''Neurodiversity and emotion''
|'''August 12th, 2025'''
|-
|2
|Post
|[https://uclearn.canberra.edu.au/courses/17386/discussion_topics/397109 Contributed to UCLearn discussion on motivation]
|
|'''August 12th, 2025'''
|-
|3
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGuided_meditation_and_emotion_regulation&amp;diff=2727092&amp;oldid=2724247 Fixed contraction]
|''Guided meditation and emotion regulation''
|'''August 12th, 2025'''
|-
|4
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&amp;diff=2727343&amp;oldid=2726931 Edited Headings to sentance case]
|''Boredom and substance use''
|'''August 12th, 2025'''
|-
|5
|Post
|[https://uclearn.canberra.edu.au/courses/17386/discussion_topics/398738 Contributed to UCLearn discussion on Work from Home]
|
|'''August 15th, 2025'''
|-
|6
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FNeurodiversity_and_emotion&amp;diff=2729206&amp;oldid=2728837 Edited Headings to sentance case]
|''Neurodiversity and emotion''
|'''August 15th, 2025'''
|-
|7
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FWork_motivation_and_self-determination_theory&amp;diff=2739753&amp;oldid=2726464 Fixed grammar mistake in Conclusion and added in key reference]
|''Work motivation and self-determination theory''
|'''August 20th, 2025'''
|-
|8
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FVulnerable_dark_triad%2C_motivation%2C_and_emotion&amp;diff=2739757&amp;oldid=2673928 Added in VDT traits to overview, fixed spelling mistake]
|''Vulnerable dark triad, motivation, and emotion''
|'''August 20th, 2025'''
|-
|9
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FDark_tetrad_and_emotion&amp;diff=2739758&amp;oldid=2581585 Fixed multiple grammar mistakes and fixed sentences]
|''Dark tetrad and emotion''
|'''August 20th, 2025'''
|-
|10
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-disclosure_motivation&amp;diff=2739771&amp;oldid=2739300 Fixed casing for headings]
|''Self-disclosure motivation''
|'''August 20th, 2025'''
|-
|11
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FNegative_affect_and_substance_use_relapse&amp;diff=2739775&amp;oldid=2739541 Flagged missing citations, fixed casing for headings]
|''Negative affect and substance use relapse''
|'''August 20th, 2025'''
|-
|12
|Comment
|[[Talk:Motivation and emotion/Book/2025/Music and social bonding#Why social connection is important to humans|Provided information and sources on why social connection is important for humans]]
|''Music and social bonding''
|'''August 21st, 2025'''
|-
|13
|Post
|[https://uclearn.canberra.edu.au/courses/17386/discussion_topics/400065 Posted a discussion page on UCLearn: &quot;Big light - on or off?&quot;]
|
|'''August 21st, 2025'''
|-
|14
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&amp;diff=2740144&amp;oldid=2739943 Adjusted references to match APA 7th, added internal links in overview]
|''Boredom and substance use''
|'''August 26th, 2025'''
|-
|15
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&amp;diff=2756247&amp;oldid=2751817 Made several changes to sentence structure, grammar/spelling]
|''Boredom and substance use''
|'''September 27th, 2025'''
|-
|16
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FLighting_and_mood&amp;diff=2766520&amp;oldid=2763226 Applied marking feedback (spelling/grammar) to my own chapter]
|''Lighting and mood''
|'''November 5th, 2025'''
|}
#

== References ==</text>
      <sha1>gx27gaaog61ds4hki03rtwlcl8kqn1x</sha1>
    </revision>
    <revision>
      <id>2834697</id>
      <parentid>2834695</parentid>
      <timestamp>2026-09-27T18:55:37Z</timestamp>
      <contributor>
        <username>Jaspershields</username>
        <id>3005613</id>
      </contributor>
      <comment>/* more? */</comment>
      <origin>2834697</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5288" sha1="fd8w61heerdggwaqvoxmgtl83125gzx" xml:space="preserve">jasper shields was previously a full time student at the [https://www.canberra.edu.au/ University of Canberra], graduating with decent grades towards the end of 2025. he moved up to the gold coast to continue his education and study psychology honours.[[File:JasperShields.jpg|thumb|'''Figure 1.''' Jasper in 2024]]
he thinks this unit was very good, and thanks dr. neill for his effort and attention in preserving the work of students.

== book chapter ==
throughout the semester, he worked on his book chapter titled &quot;[[Motivation and emotion/Book/2025/Lighting and mood|Lighting and mood]]&quot;. 

it's worth mentioning that the current version of the chapter has been edited post marking to fix grammar and spelling mistakes, but is otherwise unchanged from submission.

== more? ==
he has a [https://www.youtube.com/channel/UChpV_jZSL9aI8ByS1ktuFiA YouTube channel,]containing some of his previous work from his time at the University of Canberra, and he is contactable on his [https://www.linkedin.com/in/jaspershields/ Linkedin]. in his free time, he plays chess, and makes levels for the free to play video game osu!. he likes to chat about psychology.

== contributions ==
{| class=&quot;wikitable sortable mw-collapsible&quot;
!#
!Action
!Description
!Book chapter
!Date
|-
|1
|Comment
|[[Talk:Motivation and emotion/Book/2025/Neurodiversity and emotion#Autism Spectrum Disorder and Emotional Regulation|Made suggestion regarding Autism Spectrum Disorder (ASD) and Emotional Regulation]]
|''Neurodiversity and emotion''
|'''August 12th, 2025'''
|-
|2
|Post
|[https://uclearn.canberra.edu.au/courses/17386/discussion_topics/397109 Contributed to UCLearn discussion on motivation]
|
|'''August 12th, 2025'''
|-
|3
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGuided_meditation_and_emotion_regulation&amp;diff=2727092&amp;oldid=2724247 Fixed contraction]
|''Guided meditation and emotion regulation''
|'''August 12th, 2025'''
|-
|4
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&amp;diff=2727343&amp;oldid=2726931 Edited Headings to sentance case]
|''Boredom and substance use''
|'''August 12th, 2025'''
|-
|5
|Post
|[https://uclearn.canberra.edu.au/courses/17386/discussion_topics/398738 Contributed to UCLearn discussion on Work from Home]
|
|'''August 15th, 2025'''
|-
|6
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FNeurodiversity_and_emotion&amp;diff=2729206&amp;oldid=2728837 Edited Headings to sentance case]
|''Neurodiversity and emotion''
|'''August 15th, 2025'''
|-
|7
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FWork_motivation_and_self-determination_theory&amp;diff=2739753&amp;oldid=2726464 Fixed grammar mistake in Conclusion and added in key reference]
|''Work motivation and self-determination theory''
|'''August 20th, 2025'''
|-
|8
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FVulnerable_dark_triad%2C_motivation%2C_and_emotion&amp;diff=2739757&amp;oldid=2673928 Added in VDT traits to overview, fixed spelling mistake]
|''Vulnerable dark triad, motivation, and emotion''
|'''August 20th, 2025'''
|-
|9
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FDark_tetrad_and_emotion&amp;diff=2739758&amp;oldid=2581585 Fixed multiple grammar mistakes and fixed sentences]
|''Dark tetrad and emotion''
|'''August 20th, 2025'''
|-
|10
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-disclosure_motivation&amp;diff=2739771&amp;oldid=2739300 Fixed casing for headings]
|''Self-disclosure motivation''
|'''August 20th, 2025'''
|-
|11
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FNegative_affect_and_substance_use_relapse&amp;diff=2739775&amp;oldid=2739541 Flagged missing citations, fixed casing for headings]
|''Negative affect and substance use relapse''
|'''August 20th, 2025'''
|-
|12
|Comment
|[[Talk:Motivation and emotion/Book/2025/Music and social bonding#Why social connection is important to humans|Provided information and sources on why social connection is important for humans]]
|''Music and social bonding''
|'''August 21st, 2025'''
|-
|13
|Post
|[https://uclearn.canberra.edu.au/courses/17386/discussion_topics/400065 Posted a discussion page on UCLearn: &quot;Big light - on or off?&quot;]
|
|'''August 21st, 2025'''
|-
|14
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&amp;diff=2740144&amp;oldid=2739943 Adjusted references to match APA 7th, added internal links in overview]
|''Boredom and substance use''
|'''August 26th, 2025'''
|-
|15
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&amp;diff=2756247&amp;oldid=2751817 Made several changes to sentence structure, grammar/spelling]
|''Boredom and substance use''
|'''September 27th, 2025'''
|-
|16
|Edit
|[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FLighting_and_mood&amp;diff=2766520&amp;oldid=2763226 Applied marking feedback (spelling/grammar) to my own chapter]
|''Lighting and mood''
|'''November 5th, 2025'''
|}
#

== References ==</text>
      <sha1>fd8w61heerdggwaqvoxmgtl83125gzx</sha1>
    </revision>
  </page>
  <page>
    <title>Category:Emotional Competency</title>
    <ns>14</ns>
    <id>323165</id>
    <revision>
      <id>2834763</id>
      <parentid>2730915</parentid>
      <timestamp>2026-09-28T01:28:45Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>added [[Category:Emotional intelligence]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <origin>2834763</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="159" sha1="k1wiqiku1ov7jykpnaxjvtgrszve4uh" xml:space="preserve">Emotional competency is the skill to recognize, interpret, and respond constructively to emotions in yourself and others.[

[[Category:Emotional intelligence]]</text>
      <sha1>k1wiqiku1ov7jykpnaxjvtgrszve4uh</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Future orientation and criminal behaviour</title>
    <ns>0</ns>
    <id>330075</id>
    <revision>
      <id>2834772</id>
      <parentid>2833305</parentid>
      <timestamp>2026-09-28T01:44:25Z</timestamp>
      <contributor>
        <username>U3279062</username>
        <id>3106301</id>
      </contributor>
      <comment>light editing across page</comment>
      <origin>2834772</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25269" sha1="26nsesyn8ex5ao5xdos47aaa2wt0lmq" xml:space="preserve">{{title|Future orientation and criminal behaviour:&lt;br&gt;How does future orientation influence the risk of criminal activity?}}

==Overview==
{{RoundBoxTop|theme=7}}

[[File:Exams during Covid-19 - France (50651621763).jpg|thumb|200px|'''Figure 1'''. Exams during Covid-19.]]

; Acar and Kaya (2024) study

Amongst 23 adolescent participants, psychological drivers, social and environmental influences, and cognitive and planning capacities were identified as primary categories influencing adolescent future orientation, alongside family, peers, educational access, and cultural norms contributing to the facilitation or hindering of future aspirations (Acar &amp; Kaya, 2024). 

'''Psychological drivers''': 'hope and optimism, fear of failure, self-identity, motivation and ambition, and emotional regulation' (Acar &amp; Kaya, 2024). 

'''Social and environmental''': 'parental support, peer influence, school environment, socioeconomic constraints, and community expectations' (Acar &amp; Kaya, 2024). 

'''Cognitive and planning capacities''': 'goal-setting skills, time perspective, decision-making ability, problem-solving skills, information-seeking behavior [''sic''], and academic self-efficacy' (Acar &amp; Kaya, 2024).     

{{RoundBoxBottom}}

When focusing on adolescents, future orientation is a vital component of their education (Johnson et al., 2014). Conversely, adolescent crime rates have been steadily decreasing (Australian Bureau of Statistics [ABS], 2026), yet they remain a major concern (Hinchcliff et al., 2026; Australian Human Rights Commission [AHRC], 2024; World Health Organisation [WHO], 2026).  

{{RoundBoxTop|theme=2}}
'''Focus questions'''

{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

'''1.''' Why is Future Orientation Important?

'''2.''' Can Future Orientation Mitigate Criminal Behaviour?  

'''3.''' Are Adolescents More Susceptible to Aversive Effects of Low Future Orientation?

'''4.''' Does Criminal Behaviour Depend Upon Future-Oriented Cognition? 
{{RoundBoxBottom}}

== Future Orientation ==
{{ic|Include at least an introductory paragraph before branching into sub-headings}}

===What is Future Orientation===
According to House et al. (2004) {{g}} future orientation is the tendency for a person to engage in thoughts, actions, and the delaying of immediate gratification in favour of future goals. In adolescents, many disciplines have affirmed the importance of future-oriented behaviour in promoting positive health and development (Johnson et al., 2016). Trommsdorff (9179) proposed that future orientation included eight dimensions: extension, detail, domain, affect, motivation, control, sequence of events, and number of cognitions. A study by Beal (2011) found that Trommsdorff's (1979) conceptualisation suggests that different influences can indicate future orientation across the eight dimensions. 

==Influences on Future Orientation==
Multidisciplinary findings have indicated that a variety of influences contribute to future orientation. Much psychological research has focused on adolescents and young adults, so findings are skewed, though still extensive and reliable.    

===Psychological Influences===
McCue et al. (2019) examined how future orientation affects adolescent decision-making. Previously, connectedness to future-self and episodic future thinking have demonstrated future-oriented cognition, and while their correlation is undetermined, individual measures have shown significant effects in minimising hazardous decision-making in both adolescents and adults (Grekin et al., 2025; Goger et al., 2024; Arora, 2023). 

Future-connectedness is defined by behaviours and cognition involving the active imagination, verbal and physical communication, and other related engagements towards a person’s future in all forms. Disconnect from the future self has been shown to increase irrational, short-sighted, and risky behaviours deemed beneficial in the present (Hershfield, 2011). Episodic future thinking is defined as the mental simulation of possible future occurrences and supports future orientation by highlighting the importance of delaying rewards and countering impulsive behaviours (Gilbert &amp; Wilson, 2007; Benoit et al., 2011; Bar, 2009). This is particularly important because future-oriented thinking has historically been shown to be essential among adolescents (Nurmi, 1991), and to reduce impulsive and risky behaviour in favour of future rewards (Steinberg, 2004; Madden &amp; Bickel, 2012).

McCue et al. (2019) found a significant positive correlation between future-connectedness and episodic future thinking, claiming to be one of, if not the first, studies to identify a beneficial relationship and independent effects within adolescents. Giollabhui et al. (2018) found that increased future-oriented behaviours and cognition contribute to reduced adolescent hopelessness. During adolescence, hopelessness is categorised by adamant negative future expectations and vulnerability (Casey et al., 2008; Lester, 2015).

The severity of hopelessness in adolescence, and its relation to increased engagement in violent and/or high-risk behaviours, severe depression and anxiety, and unsafe sexual conduct (Becker-Weidman et al., 2009; Bolland, 2003; James et al., 2017; Giollabhui et al., 2018), indicates the importance of minimising adolescent hopelessness. Suicide has been a primary cause of death amongst adolescents for many years, skyrocketing after COVID-19 (Kim et al., 2024), and highlighting the worrisome trend. Considering the relationship between hopelessness, future-orientation, and adolescent depression and suicidal behaviours, increasing positive cognitions actively declines time-limited rates of hopelessness across confounds, including socio-economic status and stressors (Giollabhui et al., 2018).

=== Future Orientation and Criminal Activity ===

In relation to criminal behaviour, higher levels of future-oriented cognition indicate greater consideration of the long-term consequences of offending (Kwon et al., 2025); by contrast, limited future orientation correlates with increased criminal behaviour by encouraging immediate gratification and reducing consideration of delayed consequences (Kübel et al., 2023). 

Clinkinbeard (2014) found that high future orientation and self-control predict adolescent criminal behaviour; particularly, higher future-oriented cognition mitigates negative effects of low self-control and lessens impulsivity, risky behaviour engagement, criminal cognition, and adolescent crime rates (Khetarpal et al., 2021; Squillaro &amp; Bixter, 2025; Simmons et al., 2019). 

Importantly, Walters (2019) identified a gradual decrease in criminal behaviour with maturation, associated with increased future orientation and decreased adolescent hopelessness. Similarly, Wood et al. (2017) recognised that cognitive immaturity relates to engagement in aversive behaviours, and thereby suffering future negative life effects; promoting future orientation and positive childhood effects can mitigate these negative life effects and promote sustainability (Kübel et al., 2023; Cauffman et al., 2023; Mueller et al., 2023). 

;Quiz
Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{A variety of influences contribute to future orientation:
|type=&quot;(+)&quot;}
+ True
- False

{Future-connectedness is defined by behaviours and cognition involving the engagement towards a person’s future:
|type=&quot;(+)&quot;}
+ True
- False

{Disconnect from the future self has been shown to have no aversive risks:
|type=&quot;(+)&quot;}
+ True
- False

{Hopelessness is a persistent and serious outcome of adolescence:
|type=&quot;(+)&quot;}
+ True
- False

{Future orientation is ''the'' primary indicator for engagement with criminal behaviour:
|type=&quot;(-)&quot;}
+ True
- False
&lt;/quiz&gt;

==Key Points==
* Future orientation is the tendency for a person to engage in thoughts and actions in favour of future goals. 
*Future-connectedness is defined by behaviours and cognition involving engagement towards a person’s future in all forms. 
*The severity of hopelessness in adolescence, and its high relation to adverse behaviours, indicates the importance of minimising adolescent hopelessness.  
* Higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending. 
* Limited future orientation correlates with increased criminal behaviour. 

=Criminal Behaviour=
Dent and Ward (2022) conceptualised criminal behaviour as ‘an action or set of actions... understood by the actor to be adaptive in relation to their functional needs, but...make it more difficult to function adaptively...to other sociocultural needs.’ Criminal behaviour can include minor crimes, such as ‘theft or traffic violations’, and major crimes, such as ‘physical abuse, sexual offense [sic], or murder’ (Prent et al., 2023), and is considered a ‘major social problem with complex causes... [and involves] a myriad of environmental, social, and psychological factors are associated with increased risk of convictions for this type of criminality’ (Prent et al., 2023; Levitt, 2012; Swaab &amp; Meynen, 2023). 

==Influences on Criminal Behaviour==
Reiterating findings from Prent et al. (2023), criminal behaviour is shaped by many complex, interconnected facets, including social, psychological, biological, and environmental factors, each with influential components such as geographical location and socio-economic status. Importantly, future orientation has shown positive effects in minimising negative behaviour across different contributing factors (Giollabhui et al., 2018).

==Psychological Influences==
In media and history, psychology has often been treated as the primary explanation for criminal acts; however, crime is not limited to a single discipline (South Australia News, 2026). While this is well established, it is often a misconception that one factor is more influential than the other. 

A longstanding theory explaining human behaviour, cognition, and reasoning is Nature v Nurture. Nurture, as a psychological construct, argues that a person’s circumstances stem from upbringing and environment rather than other factors. Hadsell (2026) stated that the ‘connection between parents and children has two pathways: biological, covering both genes and hormones, and environmental.’

==Biological Influences==
When discussing the Nature concept, it is theorised that an individual's biological makeup is the sole predictor of behaviour (Alper, 1995). Baschetti (2008) argued that genetic evidence proved Nature over Nurture. Drawing on Charles Darwin’s theories of evolution and biology, current and overwhelming research indicates that criminality is heritable, yet persistent (Baschetti, 2008; Pizzi, 2004). It should be noted that Nature can refer to the genetic contribution within mental illness. 

Rao (2007) argued that all humans have a ‘dark side that loves crime and violence... [while] we may all deny it, but the contrary is true...regardless of our age, caste, social status, region, religion or education.’ [''APA style: Direct quotes should use double quotation marks and include page numbers''] Relevant to crime rates, it can be reasoned that modern-day society is the ‘most violent culture in history in the number of crimes and...the nature of brutality.’ Ling et al. (2019) explored how psychophysiology, brain mechanisms, and genetics stimulate criminal behaviour. Particularly, disrupted prefrontal cortex-amygdala connectivity and changes in autonomic functioning damaging the hypothalamic-pituitary-adrenal axis correlate with increased criminal and anti-social tendencies (Gillespie et al.,  2018; Critchley, 2005; Brzozowski &amp; Mitchell, 2018) [''APA style: Arrange multiple citations in alphabetical order''].

Baschetti (2008) signified the importance of systematic changes acknowledging the existence of a biological influence to encourage appropriate reforms and minimise negative outcomes.

==Relationship Between Criminal Behaviour and Future Orientation==
{{expand}}

==Does Future Orientation Influence the Risk of Criminal Activity?==
Future orientation is an incredibly [''Avoid overly emotive language''] important skill, particularly for adolescents or those experiencing hopelessness. A lack of future-oriented cognition can exacerbate impulsive and risky behaviour and may lead to criminal behaviour (Squillaro &amp; Bixter, 2025; Mueller et al., 2023). Given adolescents' vulnerability to criminal behaviours (Bui &amp; Deakin, 2021) and hopelessness (Bolland, 2003; Tonkus et al., 2022), encouraging future orientation is highly relevant. Reiterating that future orientation positively relates to curbing adolescent criminal activity and juvenile incarceration (Walters, 2019; Simmons et al., 2019) reinforces future orientation as an imperative protective factor across the lifespan (Yang et al., 2026; Clinkinbeard, 2014).  

==Book Chapter Direction==
* Environmental influences in criminal activity. 
*Social influences in criminal activity. 
*Understanding the relationship between future orientation and criminal activity. 
*Protective factors within future orientation
*Risk factors within future orientation
*Future research directions

==Conclusion==
''''' Include key sentence on the importance of understanding future orientation in criminal behaviour''.'''

Currently, there is substantial research into the effects and relationships of future orientation and adolescent criminal activity. However, conclusive findings on the relevance of using future orientation to mitigate criminal behaviour remain limited. Given the strong positive correlations between future-oriented thinking and reduced disturbances in life quality, it would be unwise to ignore how future orientation can help prevent rising crime rates among adolescents ''and'' adults. 

==See also==
{{expand}}

==References==
{{Hanging indent|1=

Alper, J. S. (1995). Biological influences on criminal behaviour: How good is the evidence? *BMJ, 310*(6975), 272–273. https://doi.org/10.1136/bmj.310.6975.272

Arora, S. (2023). *“It’s a future-me problem”: How future-self continuity and message framing affect climate change risk perceptions* [Doctoral dissertation, University of Guelph]. Atrium. https://atrium.lib.uoguelph.ca/items/4d97286e-5635-45b6-a1c3-e9be464b1e7c

Bar, M. (2009). The proactive brain: Memory for predictions. *Philosophical Transactions of the Royal Society B: Biological Sciences, 364*(1521), 1235–1243. https://doi.org/10.1098/rstb.2008.0310

Baschetti, R. (2008). Genetic evidence that Darwin was right about criminality: Nature, not nurture. *Medical Hypotheses, 70*(6), 1092–1102. https://doi.org/10.1016/j.mehy.2008.01.005

Becker-Weidman, E. G., Jacobs, R. H., Reinecke, M. A., Silva, S. G., &amp; March, J. S. (2010). Social problem-solving among adolescents treated for depression. *Behaviour Research and Therapy, 48*(1), 11–18. https://doi.org/10.1016/j.brat.2009.08.006

Benoit, R. G., Gilbert, S. J., &amp; Burgess, P. W. (2011). A neural mechanism mediating the impact of episodic prospection on farsighted decisions. *The Journal of Neuroscience, 31*(18), 6771–6779. https://doi.org/10.1523/JNEUROSCI.6559-10.2011

Bickel, W. K., Athamneh, L. N., Basso, J. C., Mellis, A. M., DeHart, W. B., Craft, W. H., &amp; Pope, D. (2019). Excessive discounting of delayed reinforcers as a trans-disease process: Update on the state of the science. *Current Opinion in Psychology, 30*, 59–64. https://doi.org/10.1016/j.copsyc.2019.01.005

Bolland, J. M. (2003). Hopelessness and risk behaviour among adolescents living in high-poverty inner-city neighbourhoods. *Journal of Adolescence, 26*, 145–158. https://doi.org/10.1016/S0140-1971(02)00136-7

Bui, L., &amp; Deakin, J. (2021). What we talk about when we talk about vulnerability and youth crime: A narrative review. *Aggression and Violent Behavior, 58*, 101605. https://doi.org/10.1016/j.avb.2021.101605

Cauffman, E., Gillespie, M. L., Beardslee, J., Davis, F., Hernandez, M., &amp; Williams, T. (2023). Adolescent contact, lasting impact? Lessons learned from two longitudinal studies spanning 20 years of developmental science research with justice-system-involved youths. *Psychological Science in the Public Interest, 24*(3), 133–161. https://doi.org/10.1177/15291006231205173

Chadwick, R. (n.d.). Genetics and crime. In M. Levitt (Ed.), *Encyclopedia of applied ethics* (2nd ed., pp. 462–469). Academic Press. https://doi.org/10.1016/B978-0-12-373932-2.000375

Clinkinbeard, S. S. (2014). What lies ahead: An exploration of future orientation, self-control, and delinquency. *Criminal Justice Review, 39*(1), 19–36. https://doi.org/10.1177/0734016813501193

Critchley, H. D. (2005). Neural mechanisms of autonomic, affective, and cognitive integration. *The Journal of Comparative Neurology, 493*(1), 154–166. https://doi.org/10.1002/cne.20749

Dent, H., &amp; Ward, T. (2022). Emotion and correctional practice: Theoretical foundations of an enactive practice framework. *Aggression and Violent Behavior, 63*, 101672. https://doi.org/10.1016/j.avb.2021.101672

Faura-Garcia, J., Calvete, E., &amp; Orue, I. (2024). Longitudinal associations between nonsuicidal self-injury, depressive symptoms, hopelessness, and emotional dysregulation in adolescents. *Archives of Suicide Research, 28*(3), 800–814. https://doi.org/10.1080/13811118.2023.2237075

Gillespie, S. M., Brzozowski, A., &amp; Mitchell, I. J. (2018). Self-regulation and aggressive antisocial behaviour: Insights from amygdala-prefrontal and heart-brain interactions. *Psychology, Crime &amp; Law, 24*(3), 243–257. https://doi.org/10.1080/1068316X.2017.1414816

Gilbert, D. T., &amp; Wilson, T. D. (2007). Prospection: Experiencing the future. *Science, 317*(5843), 1351–1354. https://doi.org/10.1126/science.1144161

Goger, P., Nam, R. J., Lowry, N., Bell, K. A., Parvez, N., Pollak, O. H., Robinaugh, D. J., Schacter, D. L., &amp; Cha, C. B. (2024). Testing the modifiability of episodic future thinking and episodic memory among suicidal and nonsuicidal adolescents. *JCPP Advances, 4*(3), e12236. https://doi.org/10.1002/jcv2.12236

Grekin, E. R., Thomas, H. A., Souweidane, M. A., &amp; Stidham, J. L. (2025). The effects of future self-continuity interventions on behavioral outcomes in adults: A systematic review of the literature. *Personality Science, 6*. https://doi.org/10.1177/27000710251391610

Hadsell, L. (2026). Parents: Nature and nurture. In *Social economies*. Palgrave Macmillan. https://doi.org/10.1007/978-3-032-21916-9_18

Hershfield, H. E. (2011). Future self-continuity: How conceptions of the future self transform intertemporal choice. *Annals of the New York Academy of Sciences, 1235*, 30–43. https://doi.org/10.1111/j.1749-6632.2011.06201.x

James, S., Reddy, S. P., Ellahebokus, A., Sewpaul, R., &amp; Naidoo, P. (2017). The association between adolescent risk behaviours and feelings of sadness or hopelessness: A cross-sectional survey of South African secondary school learners. *Psychology, Health &amp; Medicine, 22*(7), 778–789. https://doi.org/10.1080/13548506.2017.1300669

Johnson, S. R., Blum, R. W., &amp; Cheng, T. L. (2014). Future orientation: A construct with implications for adolescent health and wellbeing. *International Journal of Adolescent Medicine and Health, 26*(4), 459–468. https://doi.org/10.1515/ijamh-2013-0333

Khetarpal, S. K., Szoko, N., Ragavan, M. I., &amp; Culyba, A. J. (2021). Future orientation as a cross-cutting protective factor against multiple forms of violence. *The Journal of Pediatrics, 235*, 288–291. https://doi.org/10.1016/j.jpeds.2021.05.019

Kübel, L. S., Deitzer, R. J., Frankenhuis, E. W., Ribeaud, D., Eisner, P. M., &amp; van Gelder, J.-L. (2023). The shortsighted victim: Short-term mindsets mediate the link between victimization and later offending. *Journal of Criminal Justice, 86*, 102062. https://doi.org/10.1016/j.jcrimjus.2023.102062

Kwon, K. Y., Sutin, A. R., &amp; Kim, J. (2025). Juvenile delinquency and cognitive function in adulthood: Differentiating violent and nonviolent behaviors and exploring multiple mechanisms. *Journal of Research on Adolescence, 35*(4), e70095. https://doi.org/10.1111/jora.70095

Lester, D. (2015). Hopelessness in adolescents. *Journal of Affective Disorders, 173*, 221–225. https://doi.org/10.1016/j.jad.2014.10.048

Ling, S., Umbach, R., &amp; Raine, A. (2019). Biological explanations of criminal behavior. *Psychology, Crime &amp; Law, 25*(6), 626–640. https://doi.org/10.1080/1068316X.2019.1572753

Mac Giollabhui, N., Hamilton, J. L., Nielsen, J., Connolly, S. L., Stange, J. P., Varga, S., Burdette, E., Olino, T. M., Abramson, L. Y., &amp; Alloy, L. B. (2018). Negative cognitive style interacts with negative life events to predict first onset of a major depressive episode in adolescence via hopelessness. *Journal of Abnormal Psychology, 127*(1), 1–11. https://doi.org/10.1037/abn0000301

McCue, R., McCormack, T., McElnay, J., Alto, A., &amp; Feeney, A. (2019). The future and me: Imagining the future and the future self in adolescent decision making. *Cognitive Development, 50*, 142–156. https://doi.org/10.1016/j.cogdev.2019.04.001

Mueller, K. C., Cavitt, J., &amp; Carey, M. T. (2023). Future orientation in justice-involved youth: The effects of adverse and positive childhood experiences, and being dual-status. *Child Abuse &amp; Neglect, 141*, 106230. https://doi.org/10.1016/j.chiabu.2023.106230

Nurmi, E. J. (1991). How do adolescents see their future? A review of the development of future orientation and planning. *Developmental Review, 11*(1), 1–59. https://doi.org/10.1016/0273-2297(91)90002-6

Peterson, M. F. (2004). [Review of the book *Culture, leadership and organizations: The GLOBE study of 62 societies*, by R. J. House, M. Javidan, P. W. Dorfman, V. Gupta, &amp; P. J. Hanges]. *Administrative Science Quarterly, 49*(4), 641–647. https://www.jstor.org/stable/4131493

Pizzi, W. J. (2004). Nature via nurture: Genes, experience, and what makes us human. *Journal of Undergraduate Neuroscience Education, 2*(2), R10–R11.

Prent, N., Jonker, A. F., Schouws, N. T. M. S., &amp; Jonker, C. (2023). The risk of criminal behavior in the elderly and patients with neurodegenerative disease. In H. Swaab &amp; G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 181–196). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00004-9

Sathyanarayana Rao, T. S. (2007). Psychiatrist and the science of criminology: Sociological, psychological and psychiatric analysis of the dark side. *Indian Journal of Psychiatry, 49*(1), 3–5. https://doi.org/10.4103/0019-5545.31511

Simmons, C., Rowan, Z., Knowles, A., Steinberg, L., Frick, P. J., &amp; Cauffman, E. (2019). A life history approach to understanding juvenile offending and aggression. *Aggression and Violent Behavior, 49*, 101317. https://doi.org/10.1016/j.avb.2019.07.012

South Australia News. (2026, April 25). *Psychology of a criminal: Why do people commit crimes?* https://sanews.in/psychology-of-a-criminal/

Squillaro, D., &amp; Bixter, M. T. (2025). The relationship between four indicators of future time orientation, criminal thinking style, and illegal behaviors. *Journal of Forensic Sciences, 70*(1), 200–214. https://doi.org/10.1111/1556-4029.15640

Steinberg, L. (2004). Risk taking in adolescence: What changes, and why? *Annals of the New York Academy of Sciences, 1021*, 51–58. https://doi.org/10.1196/annals.1308.005

Swaab, H., &amp; Meynen, G. (2023). Introduction: On brain and crime. In H. Swaab &amp; G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 3–9). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00018-9

Turton, H., Berry, K., Danquah, A., &amp; Pratt, D. (2021). The relationship between emotion dysregulation and suicide ideation and behaviour: A systematic review. *Journal of Affective Disorders Reports, 5*, 100136. https://doi.org/10.1016/j.jadr.2021.100136

Walters, G. D. (2019). Maturing out of delinquency: Unpacking the effects of identity achievement and future orientation on crime desistance. *Self and Identity, 18*(3), 267–283. https://doi.org/10.1080/15298868.2018.1439848

Wager, T. D., Waugh, C. E., Lindquist, M., Noll, D. C., Fredrickson, B. L., &amp; Taylor, S. F. (2009). Brain mediators of cardiovascular responses to social threat: Part I: Reciprocal dorsal and ventral sub-regions of the medial prefrontal cortex and heart-rate reactivity. *NeuroImage, 47*(3), 821–835. https://doi.org/10.1016/j.neuroimage.2009.05.043

Wood, D., Crapnell, T., Lau, L., et al. (2017). Emerging adulthood as a critical stage in the life course. In N. Halfon, C. B. Forrest, R. M. Lerner, et al. (Eds.), *Handbook of life course health development*. Springer. https://www.ncbi.nlm.nih.gov/books/NBK543712/

Yang, E. S., Kim, S., Oh, S.-K., Kang, H., &amp; Chung, I.-J. (2026). Disrupting the link between exposure to violence and adolescent delinquency: The protective role of expectations about the future. *Journal of Child and Family Studies*. https://doi.org/10.1007/s10826-026-03311-3

}}

==External links==
{{expand}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Cognitive]]
[[Category:Motivation and emotion/Book/Forensic]]</text>
      <sha1>26nsesyn8ex5ao5xdos47aaa2wt0lmq</sha1>
    </revision>
  </page>
  <page>
    <title>Igbo regalia and headdresses</title>
    <ns>0</ns>
    <id>330325</id>
    <revision>
      <id>2834694</id>
      <parentid>2816816</parentid>
      <timestamp>2026-09-27T18:46:18Z</timestamp>
      <contributor>
        <username>Okechiks</username>
        <id>3111718</id>
      </contributor>
      <minor />
      <comment>Changed ichafu to gele. Wrong use of words and it's meaning were changed. Misappropriation.</comment>
      <origin>2834694</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="34714" sha1="jg0tvbual4ha9pipyaf8xi34xjk5uyb" xml:space="preserve">{{Wikipedia|w:Igbo regalia and Headdresses}}

{{course}}
== Introduction ==
'''Igbo regalia and headdresses''' comprise the ceremonial headgear, hairstyles, adornments, accessories, clothing and insignia traditionally associated with the Igbo people. These include mainly the ichafu (head ties), aka (beads), helmet and okpu (caps), Other significant headdresses and regalia include: akupe (hand fans), nza (flywhisks), ofo (elephant tusks), nkpara (staffs ), ugo (eagle's feathers), wigs and other symbolic objects used to express cultural identity, social status, title, spiritual significance and ceremonial functions.&lt;ref name=&quot;:0&quot;&gt;{{Cite book |last=Shaw |first=Thurstan |title=Unearthing Igbo-Ukwu: archaeological discoveries in eastern Nigeria |date=1977 |publisher=Oxford University Press |isbn=978-0-19-575251-9 |location=Ibadan, Nigeria ; New York}}&lt;/ref&gt;&lt;ref name=&quot;:11&quot;&gt;{{Cite book |last=Church Missionary Society |url=https://babel.hathitrust.org/cgi/pt?id=ien.35556041684267&amp;seq=160 |title=Dictionary of the Ibo Language: English-Ibo |publisher=Church Missionary Society Bookshop |year=1923 |location=Lagos |pages=160}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal |last=Dike |first=P. Chike |date=1987 |title=Art, Symbol and Authority Among the Aro of South-east Nigeria |journal=Nigerian Magazine |volume=55 |pages=30-35}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot;&gt;{{Cite book |last=Poynor |first=Robin |url=http://archive.org/details/africanartatharn0000poyn |title=African art at the Harn Museum : spirit eyes, human hands |date=1995 |publisher=Gainesville : University of Florida |others=Internet Archive |isbn=978-0-8130-1325-1 |pages=115}}&lt;/ref&gt; Many forms of Igbo regalia are associated with rulers, chiefs, and titled individuals. They are used to mark rank, status, office and authority, and remain an important part of [[Igbo culture|Igbo cultural]] traditions.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Archeological excavations at Igbo-Ukwu by Charles Thurstan Shaw uncovered 9th-century burials containing ceremonial regalia, including beads, Headdresses, crowns, fly-whisks, anklets, and other prestige objects. &lt;ref name=&quot;:0&quot; /&gt;These discoveries show the long-standing use of adornments and regalia among the Igbo people.

== Learning Objectives ==
Before studying this, you should be able to:
* Differentiate between the cultural roles, historical origins, and gender associations of various Igbo headdresses and items of regalia.
* Understand the socio-political significance of specific insignia like the Okpu ozo, Okpu agu, and Akupe.
* Describe the material culture, artistic elements, and components of traditional Igbo clothing ensembles for both men and women.


== Module 1: Women's Headdresses, Coiffures, and Crowns ==

=== Gẹ̀lẹ̀ headdress ===
[[File:Beauty Queen Bianca.jpg|alt=Bianca Ojukwu wearing Ichafu|thumb|Bianca Odumegwu Ojukwu, an Igbo woman and Nigerian Minister of Foreign Affairs wearing '''GẸ̀LẸ̀''' headdress.]] [[File:Igbo woman styled in Igbo Ichafu (headscarf) and Akwete obiakwa. Stunning.jpg|thumb|Igbo woman wearing '''GẸ̀LẸ̀''' headdress boldly and stylishly tied with a damask head-tie fabric]] [[File:Eze Obi Ossai and wives wearing Ichafu with wrappers and elephant tusk Ivory (Odu).jpg|alt=An image of Igbo women in 1841 wearing Ichafu headdress, Odu ukwu, and wrappers and carrying Akupe while the Obi carries mkpara muo.|thumb|Igbo women described by [[William Allen (Royal Navy officer)|William Allen]] in his 1841 book as Eze Obi's wives. The image depicts them wearing headcloths known as Ichafu among the Igbo, and elephant ivory anklets known as ''odu''. They carry ''Akupe'' (handfans) while the Obi carries a staff known as ''nkpara mmuo'']] '''GẸ̀LẸ̀''' of the Yoruba people of South Western Nigeria. The GẸ̀LẸ̀ as seen above is a more elaborate mode of tying a headtie usually more voluminous than the ichafu. The GẸ̀LẸ̀ requires atleast two yards of fabric to tie on the head unlike the ichafu that requires one yard of fabric to tie on the head.

'''GẸ̀LẸ̀''' be tied with the Ankara, Aso-ofi, adire, damask fabrics, etc. In the mid 90's, the '''GẸ̀LẸ̀''' design was introduced to the Igbos through movement from the south eastern Nigeria to the Western Nigeria. Yoruba fashion was considered trending fashion which eventually have ride to Igbos adopting the '''GẸ̀LẸ̀''' 

'''Ichafu''' (also recorded as Ichafo and Icafo in historical sources) is a traditional Igbo women's headdress fashioned from fabric folded simply and not elaborate sitting on the head. It is a prominent part of ceremonial dresses typically worn by igbo women with traditional outfits. Chimamanda Ngozi Adichie in a piece republished in Style and Substance: Why What We Wear Matters, edited by Gay Garnet narrated watching her mother arrange Ichafu on her head until it sat on her head like a large flower&lt;ref name=&quot;:17&quot;&gt;{{Cite book |title=Style and substance: why what we wear matters |date=2023 |publisher=John Murray |isbn=978-1-3998-1246-7 |editor-last=Garnett |editor-first=Bay |location=London}}&lt;/ref&gt; &lt;ref&gt;{{Cite book |last=Butticci |first=Annalisa |title=African Pentecostals in Catholic Europe: the politics of presence in the twenty-first century |date=2016 |publisher=Harvard University Press |isbn=978-0-674-73709-9 |location=Cambridge, Massachusetts}}&lt;/ref&gt;Likewise, Uzo Aduba in her memoir The road is Good described wearing an elaborate Ichafu on her head in the most gorgeous coral colour.&lt;ref name=&quot;:18&quot;&gt;{{Cite book |last=Aduba |first=Uzo |title=The road is good: how a mother's strength became a daughter's purpose |date=2024 |publisher=Viking |isbn=978-0-593-29912-8 |location=New York}}&lt;/ref&gt; '''GẸ̀LẸ̀''' has influenced contemporay fashion design beyond [[Nigeria]]. A review in the [[fashion]] section of the British newspaper The Observer of Priya Ahluwalia's Spring/Summer 2026 collection noted that the designer's Jacquard knits drew inspiration from Nigerian  headwraps alongside Bollywood motifs&lt;ref&gt;{{Cite web |last=Deaman |first=Jo Jones, Helen Seamons, Sam |title=Hometown glory: highlights from London Fashion Week sprin... |url=https://observer.co.uk/style/fashion/article/hometown-glory-highlights-from-london-fashion-week-springsummer-2026 |access-date=2026-06-20 |website=The Observer |language=en}}&lt;/ref&gt; '''GẸ̀LẸ̀''' is also included in an Oxford University Press International Baccalaureate Visual Arts curriculum under the theme &quot;Textiles and cultural signs&quot; alongside Japanese kimonos.&lt;ref&gt;{{Cite book |last=Oxford |url=http://archive.org/details/visual-arts-paterson-poppy-and-vaughn-oxford-2017 |title=Visual Arts Paterson, Poppy And Vaughn Oxford 2017 Textbook |date=2017}}&lt;/ref&gt; Headcloths formed an important part of traditional Igbo women's headdresses and attire. Historical and linguistic sources record terms such as Ichafu, Icafo, Ichafo, Ichafu isi in reference to headcloths, head ties, headgears and headdresses worn by Igbo women in both everyday and ceremonial settings. This headdress was associated with markets, meetings, religious observances, weddings, celebrations, and other social occasions.&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:12&quot;&gt;{{Cite book |last=Cowen |first=Rhoda |url=https://archive.org/details/goldsilverthread0000cowe |title=The Gold and Silver Threads: A memoir of Life in the Twentieth Century |publisher=Alan Sutton Publishing Ltd |year=1994 |location=Stroud, Gloucestershire, England |publication-date=1994 |pages=67, 84}}&lt;/ref&gt;&lt;ref name=&quot;:13&quot;&gt;{{Cite book |last=Green |first=M M |url=https://archive.org/detailsibovillageaffair0000gree/page/218/mode/1up |title=Ibo Village Affairs |date=1947 |publisher=Sidgwick and Jackson |year=1947 |location=Sidgwick London |publication-date=1947 |pages=136, 218, 226}}&lt;/ref&gt; Headcloths are part of a long-standing tradition of Igbo women's headdresses and full clothing ensemble. In the ethnographic work of the anthropologist M.M. Green in Igboland, he documented women in 1947 wearing festive headcloths to market and observed that headcloths were commonly worn at meetings and social gatherings as part of their clothing ensemble.&lt;ref name=&quot;:13&quot; /&gt; Igbo headtie is worn in a simple way usually with one yard fabrics. Writing about her experiences in Igboland, Rhoda Cowen described Igbo headtie referred to as Ichafo and Icafo as a simple brightly colored headdress worn in a simple and stylish manner noting that some were wrapped around the head with projecting ends, while others formed striking elements of women's attire alongside vibrant colored fabrics or textiles and gold jewellery.&lt;ref name=&quot;:12&quot; /&gt; Igbo headtie has also been recorded in ethnographic studies of Igbo dress as a headgear fashioned from a decorative piece of cloth worn around the head known as Ichafo. It is fashioned from a piece of cloth approximately six by three feet in size, folded into triangular or rectangular forms and wrapped around the head in an elevated manner that creates a style resembling a tall turban with decorative bow. According to the author, a headgear is distinguished from a head-tie which is smaller and worn flat around the head in the Nigerian context while defining Igbo style of headtie as a headgear.&lt;ref name=&quot;:9&quot; /&gt;

[[File:Girl on ichafu.jpg|alt=A girl wearing Ịchafú n'isi|thumb|An Igbo girl wearing '''GẸ̀LẸ̀''' headdress]]


=== Helmet coiffures and beaded crowns ===
Igbo headdresses also include elaborate female head adornments, as well as decorative headgear featuring elaborate coiffures worn by masquerades such as the ''Agbogho Mmuo'' (maiden spirit) during festivals and cultural events. The Headdresses and costumes are intended to depict female figures and their feminine appearance and attributes.&lt;ref&gt;{{Cite book |last=Celenko |first=Theodore |title=A treasury of African art from the Harrison Eiteljorg Collection |last2=Eiteljorg |first2=Harrison |date=1983 |publisher=Indiana University Press |isbn=978-0-253-11057-2 |location=Bloomington}}&lt;/ref&gt; [[File:Ancient Igbo helmet Coiffures and headgear (1921).jpg|alt=Helmet Coiffures and Headgears of Ancient Igbo brides|thumb|Ethnographic photos of Igbo brides known as Nkpu brides of prospective chiefs dressed in helmet Coiffures and heagears, necklaces of Leapard teeth and aggry beads 1920.]] Helmet-shaped coiffures were among the elaborate hairstyles historically worn by Igbo women during courtship, marriage festivities, and other ceremonial occasions. Some were built on a foundation of clay, charcoal and palm oil and moulded into a crest resembling the central ridge of a Roman helmet, extending from the forehead to the nape of the neck. The coiffures were often further decorated with beads, small could plaits, cowry shells, mother-of-pearl, brass ornaments, and mirrors sewn into the hair.&lt;ref name=&quot;:6&quot;&gt;{{Cite book |last=Basden |first=George Thomas |url=https://doi.org/10.5479/sil.115290.39088000476515 |title=Among the Ibos of Nigeria |date=1921 |publisher=Seeley, Service &amp; Co., ltd}}&lt;/ref&gt;&lt;ref name=&quot;:7&quot;&gt;{{Cite journal |last=Chudi-Duru |first=Chika C. |date=2024 |title=MMA NWANYI BU EKIKE |url=https://www.journals.ezenwaohaetorc.org/index.php/UJOCC/article/viewFile/3950/4060 |journal=Ohazurume: Unizik Journal of Culture and Civilization (often abbreviated UJOCC) |volume=3 |pages=96-115}}&lt;/ref&gt; Other recorded styles include a raised helmet-like ridge formed on a clay foundation and decorated with beads, cowry shells, leopard claws, camwood paste, and other adornments. Such coiffures could signify age, status, wealth or other stages of life and formed part of ceremonial female adornment in parts of Igboland.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt; [[File:An Igbo bride adorn with the isi agu and the red bead.jpg|alt=An attire featuring Igbo beaded head crown called ngala, aka, and nza|thumb|An Igbo bride adorned with beaded crown called ''ngala'', aka(coral beads) worn around the neck and wrist and worn as earrings and carrying the ''nza'' (flywhisks) regalia.]] While elaborate Helmet-shaped coiffures were a prominent part of historical Igbo women's ceremonial adornment, contemporary ceremonial attire incorporates beaded crowns and bead-based headpieces. Studies of present-day Igbo dress culture depict the brides and her maidens wearing beaded crowns as part of the traditional attire, or decorate their hair with bead accessories during weddings and Cultural celebrations. Beaded crowns, together with corals and other ornamental beads remain a prominent feature of Igbo regalia and are associated with beauty, femininity, cultural identity, fertility, spiritual well-being and marital blessings.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;


== Module 2: Elite Signifiers, Beads, and Men's Okpu ==

=== Aka ===
[[File:Igbo Bride during her traditional marriage 20220216.jpg|alt=Igbo bride adorned with beaded accessories and Uli body arts and carrying a Calabash.|thumb|An Igbo bride dressed for ''Igbankwu'' (traditional wedding) adorned in beaded accessories featuring aka attached on the head, ''nkalari'' or ''erulu'' (coral beads) around the neck, wrists ankles and also worn as earrings. She carries the ''nza'' (fly whisk) regalia and a Calabash with her legs designed in Uli body arts.]] Beaded accessories made of glass beads are known as ''aka'' among the Igbo people. They're of various types which consist of coral beads known as ''erulu'' or ''aka'' and waist beads known as ''mgbaji''. The large coral beads are known as ''nkalari''. The ''mgbaji'' is usually a kind of flat circular coral beads worn around the waist. The large coral beads are worn around the neck, and also worn as earrings and on the wrists.&lt;ref name=&quot;:9&quot;&gt;{{Cite book |last=Melie |first=Edith E. |url=https://books.google.com/books?id=iqZbAAAAMAAJ |title=The Ozo Title of Onitsha: A Study of it's Dress and Insignia |publisher=University of Wisconsin-Madison |year=1977 |location=Madison, Wisconsin, USA |publication-date=1977 |pages=87-140}}&lt;/ref&gt;&lt;ref name=&quot;:19&quot;&gt;{{Cite book |url=https://books.google.com/books?id=UAgOAQAAMAAJ |title=Ikenga |date=1985 |publisher=Institute of African Studies, University of Nigeria. |language=en}}&lt;/ref&gt; Aka is an important part of both Igbo men's and Women's dress fashion. Among the men, it is also regarded as a status symbol across [[Africa]] and a ceremonial adornment like bridal attire for Igbo women. Gold beads are also incorporated in the dress attire.&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt; [[File:Glass beads from Igbo-Ukwu.jpg|thumb|9th Century carbon-dated Igbo-ukwu glass beads.]] Beads are highly valued in Igboland&lt;ref&gt;{{Cite book |last=Afigbo |first=A. E. |title=The Igbo and their neighbours: inter-group relations in southeastern Nigeria to 1953 |date=1987 |publisher=University Press |isbn=978-0-19-575713-2 |location=Ibadan}}&lt;/ref&gt; and have long been an important part of Igbo ceremonial dress and adornment. Excavations at Igbo-ukwu uncovered large quantities of glass and carnelian beads used in necklaces, armlets, wristlets, girdles, and other ornaments. In one royal burial, hundreds of beads were found around the skull, which suggests that the deceased wore a beaded headdress, while strings of beads and a [[copper]] crown was part of the ceremonial regalia. These discoveries dated as early as the 9th century by Thurstan Shaw show that beaded accessories was widely used as a regalia symbol and adornment in ancient Igbo society.&lt;ref name=&quot;:0&quot; /&gt;

=== Okpu ===
Okpu refers to traditional caps or helmets used by Igbo adult males of various statuses and ranks in the society for symbolic purposes. There are different kinds of Okpu worn by adult males in Igboland. The most prominent are feathered red cap known as ''okpu ozo'' and leopard cap known as ''Okpu agu''.&lt;ref name=&quot;:2&quot;&gt;{{Cite book |last=Oriji |first=John Nwachimereze |url=http://archive.org/details/ngwahistorystudy0055orij |title=Ngwa history : a study of social and economic changes in Igbo mini-states in time perspective |date=1991 |publisher=New York : P. Lang |others=Internet Archive |isbn=978-0-8204-1411-9}}&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;{{Cite book |last=M. Angulu Onwuejeogwu |url=http://archive.org/details/an-igbo-civilization-nri-kingdom-and-hegemony |title=An Igbo Civilization: Nri Kingdom and Hegemony |date=1980}}&lt;/ref&gt;

==== Okpu ozo ====
[[File:Red Cap Chiefs at an Igbo Traditional Ceremony.jpg|alt=Ndi Nze na ozo you Okpu ozo ma jidekwa Akupe, ofo, na mkpara|thumb|Titled Igbo men known as Ndi Nze na ozo wearing their traditional regalia featuring Okpu ozo (red caps made of hide with eagle's feathers). Their attire also features other Igbo regalia insignia such as Akupe, ''ofo'', ''nza'', ''nkpara'']] Okpu ozo is described as the feathered red cap worn by titled men known as Ndi Nze na Ozo. As a paraphernalia of office, okpu is regarded as sacred thereby prohibiting ordinary people from touching them.&lt;ref&gt;{{Cite book |last=Oriji |first=J. |url=https://books.google.com/books?id=WZliAQAAQBAJ |title=Political Organization in Nigeria since the Late Stone Age: A History of the Igbo People |date=2011-01-17 |publisher=Springer |isbn=978-0-230-11668-9 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:10&quot;&gt;{{Cite book |url=https://books.google.com/books?id=e-Y6dSeFt_sC |title=Ikenga |date=1980 |publisher=Institute of African Studies, University of Nigeria. |language=en}}&lt;/ref&gt;It is a high-crowned red cap made of hide and usually encircled with eight eagle feather plumes called ''ugo''.This is particularly worn by the title holders called ''Nze''. It can also be modest or low-crowned without the feather decorations worn by the ''Ozo'' title holders.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal |last=Ubani |first=Kenneth |date=2019 |title=Igbo Leadership Through the Visual Arts: Back to the Future |url= |journal=Canadian Social Science |volume=15 |issue=7}}&lt;/ref&gt; It is also known as Okpu ''mmee mmee'' which literally translates to colour of the cap and okpu ''mmee'' which signifies true loyalty. Some Igbo regions also refer to it as ''okpu eze''. &lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Ifemesia |first=C. C. |url=http://archive.org/details/traditionalhuman00ifem |title=Traditional humane living among the Igbo : an historical perspecitve |date=1979 |publisher=Enugu, Nigeria : Fourth Dimension Publishers |others=Internet Archive |isbn=978-978-156-062-0}}&lt;/ref&gt;&lt;ref name=&quot;:3&quot; /&gt; Okpu ozo is regarded as an insignia that protects the wearer when he was away from his lineage or village as well as other travellers.&lt;ref name=&quot;:4&quot; /&gt; The wearers are known as Ndi nze na ozo or red cap chiefs. To obtain the title, a candidate traditionally applied to the red cap chiefs, who supervised the initiation ceremony. During the capping ceremony, the initiate was presented with a red cap and feather as a symbol of his new status and a symbol of authority to enable him perform his duties. This capping stage is also known as ''ikube-okpu''. He also receives other insignia.&lt;ref name=&quot;:14&quot;&gt;{{Cite book |last=Umeasiegbu |first=Rems N. |title=The way we lived: Ibo customs and stories |date=1981 |publisher=Heinemann |isbn=978-0-435-90061-8 |edition=Repr |series=African writers series |location=London}}&lt;/ref&gt;

==== Okpu agu ====
[[File:Igbo kwenu.jpg|alt=Nwoke yi ekike Okpu agu na agba egwu Ohafia|thumb|Ohafia war dance performer wearing Okpu agu]] [[File:Ohafia Igbo Dance Performance Chicago.jpg|alt=Ndi yi Okpu agu agba egwu Ohafia na Chicago|thumb|Ohafia Igbo dance performance in Chicago [[United States]] featuring the Okpu agu Igbo regalia]] Some descriptions of Igbo headdresses highlight that three kinds of helmets were worn: thick ones made of coco-yam stalks, or of the bark of the Achi tree, and fine looking but thin ones made of young palm leaves or raffia hats called ''okpu uturu.''&lt;ref name=&quot;:5&quot;&gt;{{Cite book |last=Talbot |first=Percy Amaury |title=The People's of Southern Nigeria: A sketch of their History, Ethnology and Languages, with an abstract of 1921 Census. |publisher=Oxford University Press, H Milford. |year=1926 |location=London |publication-date=1926 |pages=413, 839}}&lt;/ref&gt; Among these headgears was the okpu agu, a cap associated with Ohafia warrior traditions, known as the leopard cap of bravery. It is round slanting cap made of a knitted wool of black, white and red stripes with a pattern that resemble leopard markings from which the cap derives its name.&lt;ref&gt;{{Cite book |last=McCall |first=John C |url=https://books.google.com/books?id=204eAQAAMAAJ |title=The Ohafia War Dance as Lived Experience |publisher=University of Michigan |year=1992 |location=Ann Arbor, Michigan |pages=37}}&lt;/ref&gt; The okpu agu was an important symbol of warrior achievement in Ohafia. According to historical accounts, the red colour of the cap was traditionally reserved for warriors who had taken heads in battle, or returned with the slain body of a strong animal like the leopard and the caps were dyed with the blood of war victims￼, while the black and white stripe evoked the leopard and it's qualities of strength, agility and martial prowess that were admired in accomplished warriors, as well as the leopard body movement that characterizes the Ohafia War Dance movement.&lt;ref&gt;{{Cite book |last=MacCall |first=John Christensen |title=Dancing histories: heuristic ethnography with the Ohafia Igbo |date=2000 |publisher=University of Michigan Press |isbn=978-0-472-11070-4 |location=Ann Arbor}}&lt;/ref&gt; Okpu agu was manufactured locally by process of bending over, tying and sowing. It is also known as ''Okpu-Aji'' by Nkanu Igbo people and ''Okpu Ojji'' by Abajah and ''Okpu oggu'' (fighting war caps)&lt;ref name=&quot;:5&quot; /&gt;￼


== Module 3: Ceremonial Utensils, Insignia, and Fly-Whisks ==

=== Akupe (hand fan) ===
[[File:Akupe handfan.webp|thumb|Igbo man carrying an Akupe with ofo symbol carved on it. The image also features him wearing ''Okpu'' ''Ozo'' and Isiagu flowing shirt and ''aka'' (coral beads) worn around his wrist.]] Akupe is a ceremonial traditional handfan of the Igbo people which forms part of the clothing ensemble of Igbo men and women. However, it isn't merely used as a [[fashion]] complement but functions as both a practical object and insignia or symbol of status within Igbo society. It is mainly made of raw leather but was also made of other materials like straw, palm products and copper historically. As a paraphernalia of office, it is mainly associated with the ''Nze na ozo'' title holders.&lt;ref name=&quot;:15&quot;&gt;{{Cite book |last=Uzochukwu |first=Sam |title=Traditional funeral poetry of the Igbo |date=2001 |publisher=Lagos University Press |isbn=978-978-017-624-2 |location=Lagos, Nigeria}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The use of akupe as a ceremonial regalia in Igbo land dates back to Igbo-ukwu archeology where copper hand fan with a handle made of wooden shaft along with other status symbols were unearthed from a royal burial chamber of whom was described by Thurstan Shaw as a high ranking titled man or royalty. These excavations which included the fan was dated 9th century&lt;ref name=&quot;:0&quot; /&gt;

==== Akupe as a ceremonial regalia and Fashion among the men ====
The Akupe serves as both a ceremonial insignia and an element of elite male in Igbo Society. It is part of the regalia of titled men like the Ozo title holders, chiefs. and elders. Typically carried at the hand, it complements other symbols of status such as the ''Okpu'', ''nkpara'', ''nza'' or ''odu enyi'' and the traditional flowing shirt known as Isiagu, &lt;ref&gt;{{Cite book |last=Omolade |first=Ajetunmobi |url=https://books.google.com/books?id=9GISAQAAIAAJ |title=Themes in Social Studies Education and Culture: A book of Readings |year=2000 |location=Nigeria |publication-date=2000 |pages=156-166}}&lt;/ref&gt;Akwete or Akwa ocha.&lt;ref name=&quot;:8&quot; /&gt; Although it is made of various materials, the titled men specifically use raw leather made handfans which is either called Akupe or ''agu''&lt;ref&gt;{{Cite book |last=Ndimele |first=Ozo-mekuri |title=Four Decades in the Study of Languages and Linguistics in Nigeria: A Festschrift for Kay Williamson |year=2003 |publication-date=2003 |pages=435}}&lt;/ref&gt; Other names generally used to refer to the ceremonial fan is ''nkuku'' and ''nzuzu'' especially among the ''Agbalanze'' titled group of Onitsha. The Akupe is designed and decorated in various ways but particularly in a thick and heavy pattern for the titled men. Their titles or names are written or carved at the surface of the handfans while their vehicle plate numbers are also designed as such for identification.&lt;ref name=&quot;:1&quot; /&gt; The fan is not merely acquired by titled men but bestowed on them during the capping ceremony where candidates are installed as Nze no ozo title holders or red cap chiefs. The final stage of the ceremony involves handing the candidate the ''Okpu nze na ozo'' and the ''Ugo'' feather, a fan and a sword among other insignia.&lt;ref name=&quot;:14&quot; /&gt; Akupe also serves as a ritual symbol in the Igbo mmanwu (masquerade) tradition, particularly in the cultural ceremony known as the [[Ijele Masquerade|Ijele]] dance where a significant personality among the dance group is known as Akupe carrier. While he is not a masquerade, he plays the prominent role of leading the ''Ijele'' with it's symbolic powerful Akupe. The disappearance of either the Akupe or it's bearer is believed to place the Ijele at risk. The Akupe bearer determines the movement of the Ijele, which moves or remains stationary according to the bearer's actions&lt;ref&gt;{{Cite journal |last=Ikemerike |first=Ikechukwu John |last2=Efuruhievwe |first2=Margaret Akpevweogene |date=2023 |title=Globalization as a Threat to Cultural Identity: A Case Study of Igba Ijele Dance Group of Awkuzu, Anambra State |url=https://ssjhis.org/wp-content/uploads/2024/02/33.-Globalization-as-a-Threat-to-Cultural-Identity-A-Case-Study-of-Igba-Ijele-Dance-Group-of-Awkuzu-Anambra-State.pdf |journal=South-South Journal of Humanities and International Studies |volume=6 |issue=1 |pages=475-496}}&lt;/ref&gt;

==== Akupe as a ceremonial regalia and fashion among the women ====
[[File:Akupe- locally made hand fan.jpg|thumb|Akupe made of decorative woolen textiles often used by women including brides and maids of honour ]] In Igbo society, both large hand fans and artistic hand-held fans have traditionally formed part of women's ceremonial presentation, particularly during weddings, festivals or public celebrations. Historical accounts of Igbo marriage and courtship customs as early as 1921, describe Igbo brides and her maids of honour carrying large fans during the marriage ceremony, particularly the ''Nkpu'' rite to cool and keep the bride refreshed after dancing during the ceremony&lt;ref name=&quot;:6&quot; /&gt; In Igbo dances and performance traditions, artistic hand-held fans are used for aesthetics and symbolic purposes where they form part of the ceremonial ensemble&lt;ref&gt;{{Cite journal |last=Obijiaku |first=Chidi |date=2023-02-27 |title=HYBRIDITY IN MODERN NIGERIAN MUSIC: THE CASE OF IGBO CHORAL ART MUSIC |url=https://journal.ru.ac.za/index.php/africanmusic/article/view/2456 |journal=African Music : Journal of the International Library of African Music |volume=11 |issue=4 |pages=25–42 |doi=10.21504/amj.v11i4.2456 |issn=2524-2741}}&lt;/ref&gt;

=== Fly-whisks (''Nza'', ''Odu ebule'') ===
[[File:Fly whisk (AM 2015.34.53-1).jpg|thumb|Fly-whisk (''Nza'' or ''Odu ebule'')]] Fly-whisks are known as ''Nza'' or ''odu ebule'' among the Igbo people. It is a traditional ceremonial accessory and symbol of distinction in Igbo society. It forms part of the regalia of titled men, chiefs, elders, and other person's of rank also integrated into the attire which includes the akupe, headgear, staff among others. Historical accounts of title taking ceremonies describe newly initiated titled men receiving ceremonial objects as part of their elevation into positions of honour and responsibility within the community such as the ''Ozo'' title.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Archeological evidence from Igbo-ukwu, dating to about the 9th century, shows the antiquity flywhisks within Igbo ceremonial culture. The excavations that uncovered a royal burial furnished with elaborate regalia and the reconstructions of the chamber depicted the high ranking individual holding a ceremonial fly-whisk among other insignia of authority. These regalia have been interpreted in studies as markets of prestige, rank, leadership, royalty, and ritual symbolisms in early Igboland.&lt;ref name=&quot;:0&quot; /&gt; Beyond functioning as an insignia among titled men, royalty and chiefs, fly-whisks and related descriptions of the regalia has been recorded as forming part of the attire of Igbo dancers and brides during festivities or ceremonies like dance performances and traditional marriage. According to Basden in 1921, ''Nkpu'' brides during their marriage ceremony carried cow's tail mounted on a leather handle, sometimes accompanied by small mirrors placed in specially carved hand held frames, including large fans which all formed part of their attire in the ceremony.&lt;ref name=&quot;:6&quot; /&gt; ''Odu enyi'' is also used in artistic and performance traditions of the Igbi people like the ''nkpokiti'' dance. Studies of Igbo oral poetry includes the fly-whisks among the symbolic paraphernalia carried by performers as emblems of their artistry. The flywhisks were described as part of the ceremonial props employed in dances, processions, and choral performance where they serve both symbolic and aesthetic functions as they express the Igbo culture.&lt;ref name=&quot;:15&quot; /&gt;


== Module 4: Traditional Dress and Ensembles ==

=== Igbo traditional dress and fashion ===
Igbo traditional dress and fashion for men typically comprise of loose cotton shirt or robe over an ankle-length wrapper, or loin cloths fashioned from various local Igbo fabrics like the Isiagu, Akwete or Akwaocha complemented with ''okpu'', ''akupe'', ''odu enyi'', ''mkpara'', ''ofo'' and adornments with ''aka'' (coral beads) worn around the neck and wrists. These regalia items proclaim status within the society. Royal robes, royal headdresses, silver sword were used to describe the attire of Igbo royalty like the Obi of Onitsha by Nzimiro and Henderson while the red cap chiefs dressed in their own special attire&lt;ref&gt;{{Cite book |last=Ebuziem |first=Cajetan E. |title=Doing Ministry in the Igbo Context: Towards an Emerging Model and Method for the Church in Africa- Foreword by Theophilus Okere |date=2011 |publisher=Peter Lang Inc., International Academic Publishers |isbn=978-1-4331-1154-9 |edition= |series=Bible and Theology in Africa |location=New York}}&lt;/ref&gt;&lt;ref name=&quot;:20&quot;&gt;{{Cite book |last=Nnoromele |first=Salome |url=https://archive.org/details/lifeamongibowome0000nnor |title=Life among the Ibo women of Nigeria |publisher=Lucent books, San Diego |year=1998 |location=San Diego |publication-date=1998}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; &lt;ref name=&quot;:16&quot; /&gt; Among the women, the typical tradition of dress consist of a pair of matching wrappers or double wrappers known as ''eregbor'' na ''ntukwasi'', a blouse called ''efe obi'' made of Akwete or George fabrics and Ichafu (also spelt Ichafo and Icafo). In ordinary circumstances according to M.M Green in 1947, the women wore short wrappers folded around the hips and reaching the knee with a headcloth while for ceremonial functions a blouse or tunic was worn together with the waist wrappers and festive headcloth &lt;ref name=&quot;:21&quot;&gt;{{Cite book |last=Lamb Holmes |first=Venice Judy |title=Nigerian Weaving |publisher=The Roxford Press |year=1981 |pages=247-280}}&lt;/ref&gt;￼&lt;ref name=&quot;:22&quot;&gt;{{Cite journal |last=Kent |first=Kate P. |last2=Eicher |first2=Joanne Bubolz |last3=Dendel |first3=Esther Warner |date=1978 |title=Nigerian Handcrafted Textiles |url=https://doi.org/10.2307/3335408 |journal=African Arts |volume=11 |issue=3 |pages=14 |doi=10.2307/3335408 |issn=0001-9933}}&lt;/ref&gt;&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; Igbo women complement their dressing with ''aka'' (beads) and Jewellries especially for festivities or ceremonial occasions like traditional marriage known as ''Igba nkwu''.The beadded accessories include ''aka'' or ''nkalari'' (coral beads) worn around the neck and wrists and used as earrings, ''mgbaji'' (flat circular waist beads) worn around the waist, and ''ola'' (iron bangles) around the ankles.&lt;ref name=&quot;:19&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt;

==== Layered Identical Double Wrappers (''Eregbor na Ntukwasi''), Blouse (''Efe obi'') and Ichafu ====
Traditionally, Igbo women's textiles like Akwete of all category are woven in pairs of identical design as well as sold in pairs. These pairs are not sewn together but worn together. The two pairs of wrappers are known as ''eregbor na ntukwasi,'' made of multicolored geometric or floral design. The first wrapper is wrapped around the waist and extends down to the ankle. The second wrapper overlaps the first from the waist to the knees, giving the wrappers a layered look. The wrappers are paired with a blouse known as ''efe obi'' which s tucked inside. This combination of wrappers and blouse are paired with head ties described by Chimamanda Adichie as Ichafu. The textiles used are usually Akwete and George.&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:21&quot; /&gt;&lt;ref name=&quot;:22&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; In recounting her past, Chimamanda Ngozi Adichie describes how she had seen her mother dress up in her double wrappers, blouse and Ichafu. ￼ 
{{Blockquote|text=She folded and twisted and pinned her Ichafu until it sat on her head like a large flower. She wrapped her George - heavy beaded cloth, alive with embroidery, always in bright shades of red or purple pink - around her waist in two layers, The first, the longer piece, hit her ankles, and the second formed an elegant tier just below her knees. Her sequinned blouse caught the light and glittered. Her shoes and handbag always matched|author=Chimamanda Ngozi Adichie|title=Style and Substance: Why What We Wear Matters}} 
This attire ensemble of layered double wrappers called ''eregbor na'' ''ntukwasi'' paired with a blouse and Ichafu (Ichafo) head tie is also complemented with jewelleries as described by Chimamanda and Uzo Aduba who had witnessed their mother and aunt dress up.&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:18&quot; /&gt; [[File:Igbo woman wearing Jooji obiakwa(double wrapper) with uweobi and Ichafu.jpg|left|thumb|A full clothing ensemble of the Igbo woman featuring ''Eregbor na ntukwasi'' (Layered identical George wrappers) paired with a puffed sleeved blouse ''(efe elu)'' and an Ichafu (Ichafo) headdress]] [[File:An igbo woman in Ichafu headdress.jpg|thumb|An Igbo woman wearing a fitted blouse (''efe obi'')]]


== See also ==
* [[Igbo culture]]
== ==

[[Category:Igbo culture]]
[[Category:Material Culture]]
[[Category:Learning Modules]]</text>
      <sha1>jg0tvbual4ha9pipyaf8xi34xjk5uyb</sha1>
    </revision>
    <revision>
      <id>2834714</id>
      <parentid>2834694</parentid>
      <timestamp>2026-09-27T20:52:39Z</timestamp>
      <contributor>
        <username>Joshey11</username>
        <id>3111726</id>
      </contributor>
      <minor />
      <comment>/* Gẹ̀lẹ̀ headdress */</comment>
      <origin>2834714</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="34702" sha1="ef4dqz0szsxy897gte2ulj9w4xdr6jz" xml:space="preserve">{{Wikipedia|w:Igbo regalia and Headdresses}}

{{course}}
== Introduction ==
'''Igbo regalia and headdresses''' comprise the ceremonial headgear, hairstyles, adornments, accessories, clothing and insignia traditionally associated with the Igbo people. These include mainly the ichafu (head ties), aka (beads), helmet and okpu (caps), Other significant headdresses and regalia include: akupe (hand fans), nza (flywhisks), ofo (elephant tusks), nkpara (staffs ), ugo (eagle's feathers), wigs and other symbolic objects used to express cultural identity, social status, title, spiritual significance and ceremonial functions.&lt;ref name=&quot;:0&quot;&gt;{{Cite book |last=Shaw |first=Thurstan |title=Unearthing Igbo-Ukwu: archaeological discoveries in eastern Nigeria |date=1977 |publisher=Oxford University Press |isbn=978-0-19-575251-9 |location=Ibadan, Nigeria ; New York}}&lt;/ref&gt;&lt;ref name=&quot;:11&quot;&gt;{{Cite book |last=Church Missionary Society |url=https://babel.hathitrust.org/cgi/pt?id=ien.35556041684267&amp;seq=160 |title=Dictionary of the Ibo Language: English-Ibo |publisher=Church Missionary Society Bookshop |year=1923 |location=Lagos |pages=160}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal |last=Dike |first=P. Chike |date=1987 |title=Art, Symbol and Authority Among the Aro of South-east Nigeria |journal=Nigerian Magazine |volume=55 |pages=30-35}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot;&gt;{{Cite book |last=Poynor |first=Robin |url=http://archive.org/details/africanartatharn0000poyn |title=African art at the Harn Museum : spirit eyes, human hands |date=1995 |publisher=Gainesville : University of Florida |others=Internet Archive |isbn=978-0-8130-1325-1 |pages=115}}&lt;/ref&gt; Many forms of Igbo regalia are associated with rulers, chiefs, and titled individuals. They are used to mark rank, status, office and authority, and remain an important part of [[Igbo culture|Igbo cultural]] traditions.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Archeological excavations at Igbo-Ukwu by Charles Thurstan Shaw uncovered 9th-century burials containing ceremonial regalia, including beads, Headdresses, crowns, fly-whisks, anklets, and other prestige objects. &lt;ref name=&quot;:0&quot; /&gt;These discoveries show the long-standing use of adornments and regalia among the Igbo people.

== Learning Objectives ==
Before studying this, you should be able to:
* Differentiate between the cultural roles, historical origins, and gender associations of various Igbo headdresses and items of regalia.
* Understand the socio-political significance of specific insignia like the Okpu ozo, Okpu agu, and Akupe.
* Describe the material culture, artistic elements, and components of traditional Igbo clothing ensembles for both men and women.


== Module 1: Women's Headdresses, Coiffures, and Crowns ==

=== Gẹ̀lẹ̀ headdress ===
[[File:Beauty Queen Bianca.jpg|alt=Bianca Ojukwu wearing Ichafu|thumb|Bianca Odumegwu Ojukwu, an Igbo woman and Nigerian Minister of Foreign Affairs wearing '''GẸ̀LẸ̀''' headdress.]] [[File:Igbo woman styled in Igbo Ichafu (headscarf) and Akwete obiakwa. Stunning.jpg|thumb|Igbo woman wearing '''GẸ̀LẸ̀''' headdress boldly and stylishly tied with a damask head-tie fabric]] [[File:Eze Obi Ossai and wives wearing Ichafu with wrappers and elephant tusk Ivory (Odu).jpg|alt=An image of Igbo women in 1841 wearing Ichafu headdress, Odu ukwu, and wrappers and carrying Akupe while the Obi carries mkpara muo.|thumb|Igbo women described by [[William Allen (Royal Navy officer)|William Allen]] in his 1841 book as Eze Obi's wives. The image depicts them wearing headcloths known as Ichafu among the Igbo, and elephant ivory anklets known as ''odu''. They carry ''Akupe'' (handfans) while the Obi carries a staff known as ''nkpara mmuo'']] '''GẸ̀LẸ̀''' of the Yoruba people of South Western Nigeria. The GẸ̀LẸ̀ as seen above is a more elaborate mode of tying a headtie usually more voluminous than the ichafu. The GẸ̀LẸ̀ requires atleast two yards of fabric to tie on the head unlike the ichafu that requires one yard of fabric to tie on the head.

'''GẸ̀LẸ̀''' be tied with the Ankara, Aso-ofi, adire, damask fabrics, etc. In the mid 90's, the '''GẸ̀LẸ̀''' design was introduced to the Igbos through movement from the south eastern Nigeria to the Western Nigeria. Yoruba fashion was considered trending fashion which eventually have ride to Igbos adopting the '''GẸ̀LẸ̀''' 

'''Ichafu''' (also recorded as Ichafo and Icafo in historical sources) is a traditional Igbo women's headdress fashioned from fabric folded simply and not elaborate sitting on the head. It is a prominent part of ceremonial dresses typically worn by igbo women with traditional outfits. Chimamanda Ngozi Adichie in a piece republished in Style and Substance: Why What We Wear Matters, edited by Gay Garnet narrated watching her mother arrange gèlè on her head until it sat on her head like a large flower&lt;ref name=&quot;:17&quot;&gt;{{Cite book |title=Style and substance: why what we wear matters |date=2023 |publisher=John Murray |isbn=978-1-3998-1246-7 |editor-last=Garnett |editor-first=Bay |location=London}}&lt;/ref&gt; &lt;ref&gt;{{Cite book |last=Butticci |first=Annalisa |title=African Pentecostals in Catholic Europe: the politics of presence in the twenty-first century |date=2016 |publisher=Harvard University Press |isbn=978-0-674-73709-9 |location=Cambridge, Massachusetts}}&lt;/ref&gt;Likewise, Uzo Aduba in her memoir The road is Good described wearing an elaborate gèlè on her head in the most gorgeous coral colour.&lt;ref name=&quot;:18&quot;&gt;{{Cite book |last=Aduba |first=Uzo |title=The road is good: how a mother's strength became a daughter's purpose |date=2024 |publisher=Viking |isbn=978-0-593-29912-8 |location=New York}}&lt;/ref&gt; '''Gèlè''' has influenced contemporay fashion design beyond [[Nigeria]]. A review in the [[fashion]] section of the British newspaper The Observer of Priya Ahluwalia's Spring/Summer 2026 collection noted that the designer's Jacquard knits drew inspiration from Nigerian  headwraps alongside Bollywood motifs&lt;ref&gt;{{Cite web |last=Deaman |first=Jo Jones, Helen Seamons, Sam |title=Hometown glory: highlights from London Fashion Week sprin... |url=https://observer.co.uk/style/fashion/article/hometown-glory-highlights-from-london-fashion-week-springsummer-2026 |access-date=2026-06-20 |website=The Observer |language=en}}&lt;/ref&gt; '''Gèlè''' is also included in an Oxford University Press International Baccalaureate Visual Arts curriculum under the theme &quot;Textiles and cultural signs&quot; alongside Japanese kimonos.&lt;ref&gt;{{Cite book |last=Oxford |url=http://archive.org/details/visual-arts-paterson-poppy-and-vaughn-oxford-2017 |title=Visual Arts Paterson, Poppy And Vaughn Oxford 2017 Textbook |date=2017}}&lt;/ref&gt; Headcloths formed an important part of traditional Igbo women's headdresses and attire. Historical and linguistic sources record terms such as Ichafu, Icafo, Ichafo, Ichafu isi in reference to headcloths, head ties, headgears and headdresses worn by Igbo women in both everyday and ceremonial settings. This headdress was associated with markets, meetings, religious observances, weddings, celebrations, and other social occasions.&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:12&quot;&gt;{{Cite book |last=Cowen |first=Rhoda |url=https://archive.org/details/goldsilverthread0000cowe |title=The Gold and Silver Threads: A memoir of Life in the Twentieth Century |publisher=Alan Sutton Publishing Ltd |year=1994 |location=Stroud, Gloucestershire, England |publication-date=1994 |pages=67, 84}}&lt;/ref&gt;&lt;ref name=&quot;:13&quot;&gt;{{Cite book |last=Green |first=M M |url=https://archive.org/detailsibovillageaffair0000gree/page/218/mode/1up |title=Ibo Village Affairs |date=1947 |publisher=Sidgwick and Jackson |year=1947 |location=Sidgwick London |publication-date=1947 |pages=136, 218, 226}}&lt;/ref&gt; Headcloths are part of a long-standing tradition of Igbo women's headdresses and full clothing ensemble. In the ethnographic work of the anthropologist M.M. Green in Igboland, he documented women in 1947 wearing festive headcloths to market and observed that headcloths were commonly worn at meetings and social gatherings as part of their clothing ensemble.&lt;ref name=&quot;:13&quot; /&gt; Igbo headtie is worn in a simple way usually with one yard fabrics. Writing about her experiences in Igboland, Rhoda Cowen described Igbo headtie referred to as Ichafo and Icafo as a simple brightly colored headdress worn in a simple and stylish manner noting that some were wrapped around the head with projecting ends, while others formed striking elements of women's attire alongside vibrant colored fabrics or textiles and gold jewellery.&lt;ref name=&quot;:12&quot; /&gt; Igbo headtie has also been recorded in ethnographic studies of Igbo dress as a headgear fashioned from a decorative piece of cloth worn around the head known as Ichafo. It is fashioned from a piece of cloth approximately six by three feet in size, folded into triangular or rectangular forms and wrapped around the head in an elevated manner that creates a style resembling a tall turban with decorative bow. According to the author, a headgear is distinguished from a head-tie which is smaller and worn flat around the head in the Nigerian context while defining Igbo style of headtie as a headgear.&lt;ref name=&quot;:9&quot; /&gt;

[[File:Girl on ichafu.jpg|alt=A girl wearing Ịchafú n'isi|thumb|An Igbo girl wearing '''GẸ̀LẸ̀''' headdress]]


=== Helmet coiffures and beaded crowns ===
Igbo headdresses also include elaborate female head adornments, as well as decorative headgear featuring elaborate coiffures worn by masquerades such as the ''Agbogho Mmuo'' (maiden spirit) during festivals and cultural events. The Headdresses and costumes are intended to depict female figures and their feminine appearance and attributes.&lt;ref&gt;{{Cite book |last=Celenko |first=Theodore |title=A treasury of African art from the Harrison Eiteljorg Collection |last2=Eiteljorg |first2=Harrison |date=1983 |publisher=Indiana University Press |isbn=978-0-253-11057-2 |location=Bloomington}}&lt;/ref&gt; [[File:Ancient Igbo helmet Coiffures and headgear (1921).jpg|alt=Helmet Coiffures and Headgears of Ancient Igbo brides|thumb|Ethnographic photos of Igbo brides known as Nkpu brides of prospective chiefs dressed in helmet Coiffures and heagears, necklaces of Leapard teeth and aggry beads 1920.]] Helmet-shaped coiffures were among the elaborate hairstyles historically worn by Igbo women during courtship, marriage festivities, and other ceremonial occasions. Some were built on a foundation of clay, charcoal and palm oil and moulded into a crest resembling the central ridge of a Roman helmet, extending from the forehead to the nape of the neck. The coiffures were often further decorated with beads, small could plaits, cowry shells, mother-of-pearl, brass ornaments, and mirrors sewn into the hair.&lt;ref name=&quot;:6&quot;&gt;{{Cite book |last=Basden |first=George Thomas |url=https://doi.org/10.5479/sil.115290.39088000476515 |title=Among the Ibos of Nigeria |date=1921 |publisher=Seeley, Service &amp; Co., ltd}}&lt;/ref&gt;&lt;ref name=&quot;:7&quot;&gt;{{Cite journal |last=Chudi-Duru |first=Chika C. |date=2024 |title=MMA NWANYI BU EKIKE |url=https://www.journals.ezenwaohaetorc.org/index.php/UJOCC/article/viewFile/3950/4060 |journal=Ohazurume: Unizik Journal of Culture and Civilization (often abbreviated UJOCC) |volume=3 |pages=96-115}}&lt;/ref&gt; Other recorded styles include a raised helmet-like ridge formed on a clay foundation and decorated with beads, cowry shells, leopard claws, camwood paste, and other adornments. Such coiffures could signify age, status, wealth or other stages of life and formed part of ceremonial female adornment in parts of Igboland.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt; [[File:An Igbo bride adorn with the isi agu and the red bead.jpg|alt=An attire featuring Igbo beaded head crown called ngala, aka, and nza|thumb|An Igbo bride adorned with beaded crown called ''ngala'', aka(coral beads) worn around the neck and wrist and worn as earrings and carrying the ''nza'' (flywhisks) regalia.]] While elaborate Helmet-shaped coiffures were a prominent part of historical Igbo women's ceremonial adornment, contemporary ceremonial attire incorporates beaded crowns and bead-based headpieces. Studies of present-day Igbo dress culture depict the brides and her maidens wearing beaded crowns as part of the traditional attire, or decorate their hair with bead accessories during weddings and Cultural celebrations. Beaded crowns, together with corals and other ornamental beads remain a prominent feature of Igbo regalia and are associated with beauty, femininity, cultural identity, fertility, spiritual well-being and marital blessings.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;


== Module 2: Elite Signifiers, Beads, and Men's Okpu ==

=== Aka ===
[[File:Igbo Bride during her traditional marriage 20220216.jpg|alt=Igbo bride adorned with beaded accessories and Uli body arts and carrying a Calabash.|thumb|An Igbo bride dressed for ''Igbankwu'' (traditional wedding) adorned in beaded accessories featuring aka attached on the head, ''nkalari'' or ''erulu'' (coral beads) around the neck, wrists ankles and also worn as earrings. She carries the ''nza'' (fly whisk) regalia and a Calabash with her legs designed in Uli body arts.]] Beaded accessories made of glass beads are known as ''aka'' among the Igbo people. They're of various types which consist of coral beads known as ''erulu'' or ''aka'' and waist beads known as ''mgbaji''. The large coral beads are known as ''nkalari''. The ''mgbaji'' is usually a kind of flat circular coral beads worn around the waist. The large coral beads are worn around the neck, and also worn as earrings and on the wrists.&lt;ref name=&quot;:9&quot;&gt;{{Cite book |last=Melie |first=Edith E. |url=https://books.google.com/books?id=iqZbAAAAMAAJ |title=The Ozo Title of Onitsha: A Study of it's Dress and Insignia |publisher=University of Wisconsin-Madison |year=1977 |location=Madison, Wisconsin, USA |publication-date=1977 |pages=87-140}}&lt;/ref&gt;&lt;ref name=&quot;:19&quot;&gt;{{Cite book |url=https://books.google.com/books?id=UAgOAQAAMAAJ |title=Ikenga |date=1985 |publisher=Institute of African Studies, University of Nigeria. |language=en}}&lt;/ref&gt; Aka is an important part of both Igbo men's and Women's dress fashion. Among the men, it is also regarded as a status symbol across [[Africa]] and a ceremonial adornment like bridal attire for Igbo women. Gold beads are also incorporated in the dress attire.&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt; [[File:Glass beads from Igbo-Ukwu.jpg|thumb|9th Century carbon-dated Igbo-ukwu glass beads.]] Beads are highly valued in Igboland&lt;ref&gt;{{Cite book |last=Afigbo |first=A. E. |title=The Igbo and their neighbours: inter-group relations in southeastern Nigeria to 1953 |date=1987 |publisher=University Press |isbn=978-0-19-575713-2 |location=Ibadan}}&lt;/ref&gt; and have long been an important part of Igbo ceremonial dress and adornment. Excavations at Igbo-ukwu uncovered large quantities of glass and carnelian beads used in necklaces, armlets, wristlets, girdles, and other ornaments. In one royal burial, hundreds of beads were found around the skull, which suggests that the deceased wore a beaded headdress, while strings of beads and a [[copper]] crown was part of the ceremonial regalia. These discoveries dated as early as the 9th century by Thurstan Shaw show that beaded accessories was widely used as a regalia symbol and adornment in ancient Igbo society.&lt;ref name=&quot;:0&quot; /&gt;

=== Okpu ===
Okpu refers to traditional caps or helmets used by Igbo adult males of various statuses and ranks in the society for symbolic purposes. There are different kinds of Okpu worn by adult males in Igboland. The most prominent are feathered red cap known as ''okpu ozo'' and leopard cap known as ''Okpu agu''.&lt;ref name=&quot;:2&quot;&gt;{{Cite book |last=Oriji |first=John Nwachimereze |url=http://archive.org/details/ngwahistorystudy0055orij |title=Ngwa history : a study of social and economic changes in Igbo mini-states in time perspective |date=1991 |publisher=New York : P. Lang |others=Internet Archive |isbn=978-0-8204-1411-9}}&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;{{Cite book |last=M. Angulu Onwuejeogwu |url=http://archive.org/details/an-igbo-civilization-nri-kingdom-and-hegemony |title=An Igbo Civilization: Nri Kingdom and Hegemony |date=1980}}&lt;/ref&gt;

==== Okpu ozo ====
[[File:Red Cap Chiefs at an Igbo Traditional Ceremony.jpg|alt=Ndi Nze na ozo you Okpu ozo ma jidekwa Akupe, ofo, na mkpara|thumb|Titled Igbo men known as Ndi Nze na ozo wearing their traditional regalia featuring Okpu ozo (red caps made of hide with eagle's feathers). Their attire also features other Igbo regalia insignia such as Akupe, ''ofo'', ''nza'', ''nkpara'']] Okpu ozo is described as the feathered red cap worn by titled men known as Ndi Nze na Ozo. As a paraphernalia of office, okpu is regarded as sacred thereby prohibiting ordinary people from touching them.&lt;ref&gt;{{Cite book |last=Oriji |first=J. |url=https://books.google.com/books?id=WZliAQAAQBAJ |title=Political Organization in Nigeria since the Late Stone Age: A History of the Igbo People |date=2011-01-17 |publisher=Springer |isbn=978-0-230-11668-9 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:10&quot;&gt;{{Cite book |url=https://books.google.com/books?id=e-Y6dSeFt_sC |title=Ikenga |date=1980 |publisher=Institute of African Studies, University of Nigeria. |language=en}}&lt;/ref&gt;It is a high-crowned red cap made of hide and usually encircled with eight eagle feather plumes called ''ugo''.This is particularly worn by the title holders called ''Nze''. It can also be modest or low-crowned without the feather decorations worn by the ''Ozo'' title holders.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal |last=Ubani |first=Kenneth |date=2019 |title=Igbo Leadership Through the Visual Arts: Back to the Future |url= |journal=Canadian Social Science |volume=15 |issue=7}}&lt;/ref&gt; It is also known as Okpu ''mmee mmee'' which literally translates to colour of the cap and okpu ''mmee'' which signifies true loyalty. Some Igbo regions also refer to it as ''okpu eze''. &lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Ifemesia |first=C. C. |url=http://archive.org/details/traditionalhuman00ifem |title=Traditional humane living among the Igbo : an historical perspecitve |date=1979 |publisher=Enugu, Nigeria : Fourth Dimension Publishers |others=Internet Archive |isbn=978-978-156-062-0}}&lt;/ref&gt;&lt;ref name=&quot;:3&quot; /&gt; Okpu ozo is regarded as an insignia that protects the wearer when he was away from his lineage or village as well as other travellers.&lt;ref name=&quot;:4&quot; /&gt; The wearers are known as Ndi nze na ozo or red cap chiefs. To obtain the title, a candidate traditionally applied to the red cap chiefs, who supervised the initiation ceremony. During the capping ceremony, the initiate was presented with a red cap and feather as a symbol of his new status and a symbol of authority to enable him perform his duties. This capping stage is also known as ''ikube-okpu''. He also receives other insignia.&lt;ref name=&quot;:14&quot;&gt;{{Cite book |last=Umeasiegbu |first=Rems N. |title=The way we lived: Ibo customs and stories |date=1981 |publisher=Heinemann |isbn=978-0-435-90061-8 |edition=Repr |series=African writers series |location=London}}&lt;/ref&gt;

==== Okpu agu ====
[[File:Igbo kwenu.jpg|alt=Nwoke yi ekike Okpu agu na agba egwu Ohafia|thumb|Ohafia war dance performer wearing Okpu agu]] [[File:Ohafia Igbo Dance Performance Chicago.jpg|alt=Ndi yi Okpu agu agba egwu Ohafia na Chicago|thumb|Ohafia Igbo dance performance in Chicago [[United States]] featuring the Okpu agu Igbo regalia]] Some descriptions of Igbo headdresses highlight that three kinds of helmets were worn: thick ones made of coco-yam stalks, or of the bark of the Achi tree, and fine looking but thin ones made of young palm leaves or raffia hats called ''okpu uturu.''&lt;ref name=&quot;:5&quot;&gt;{{Cite book |last=Talbot |first=Percy Amaury |title=The People's of Southern Nigeria: A sketch of their History, Ethnology and Languages, with an abstract of 1921 Census. |publisher=Oxford University Press, H Milford. |year=1926 |location=London |publication-date=1926 |pages=413, 839}}&lt;/ref&gt; Among these headgears was the okpu agu, a cap associated with Ohafia warrior traditions, known as the leopard cap of bravery. It is round slanting cap made of a knitted wool of black, white and red stripes with a pattern that resemble leopard markings from which the cap derives its name.&lt;ref&gt;{{Cite book |last=McCall |first=John C |url=https://books.google.com/books?id=204eAQAAMAAJ |title=The Ohafia War Dance as Lived Experience |publisher=University of Michigan |year=1992 |location=Ann Arbor, Michigan |pages=37}}&lt;/ref&gt; The okpu agu was an important symbol of warrior achievement in Ohafia. According to historical accounts, the red colour of the cap was traditionally reserved for warriors who had taken heads in battle, or returned with the slain body of a strong animal like the leopard and the caps were dyed with the blood of war victims￼, while the black and white stripe evoked the leopard and it's qualities of strength, agility and martial prowess that were admired in accomplished warriors, as well as the leopard body movement that characterizes the Ohafia War Dance movement.&lt;ref&gt;{{Cite book |last=MacCall |first=John Christensen |title=Dancing histories: heuristic ethnography with the Ohafia Igbo |date=2000 |publisher=University of Michigan Press |isbn=978-0-472-11070-4 |location=Ann Arbor}}&lt;/ref&gt; Okpu agu was manufactured locally by process of bending over, tying and sowing. It is also known as ''Okpu-Aji'' by Nkanu Igbo people and ''Okpu Ojji'' by Abajah and ''Okpu oggu'' (fighting war caps)&lt;ref name=&quot;:5&quot; /&gt;￼


== Module 3: Ceremonial Utensils, Insignia, and Fly-Whisks ==

=== Akupe (hand fan) ===
[[File:Akupe handfan.webp|thumb|Igbo man carrying an Akupe with ofo symbol carved on it. The image also features him wearing ''Okpu'' ''Ozo'' and Isiagu flowing shirt and ''aka'' (coral beads) worn around his wrist.]] Akupe is a ceremonial traditional handfan of the Igbo people which forms part of the clothing ensemble of Igbo men and women. However, it isn't merely used as a [[fashion]] complement but functions as both a practical object and insignia or symbol of status within Igbo society. It is mainly made of raw leather but was also made of other materials like straw, palm products and copper historically. As a paraphernalia of office, it is mainly associated with the ''Nze na ozo'' title holders.&lt;ref name=&quot;:15&quot;&gt;{{Cite book |last=Uzochukwu |first=Sam |title=Traditional funeral poetry of the Igbo |date=2001 |publisher=Lagos University Press |isbn=978-978-017-624-2 |location=Lagos, Nigeria}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The use of akupe as a ceremonial regalia in Igbo land dates back to Igbo-ukwu archeology where copper hand fan with a handle made of wooden shaft along with other status symbols were unearthed from a royal burial chamber of whom was described by Thurstan Shaw as a high ranking titled man or royalty. These excavations which included the fan was dated 9th century&lt;ref name=&quot;:0&quot; /&gt;

==== Akupe as a ceremonial regalia and Fashion among the men ====
The Akupe serves as both a ceremonial insignia and an element of elite male in Igbo Society. It is part of the regalia of titled men like the Ozo title holders, chiefs. and elders. Typically carried at the hand, it complements other symbols of status such as the ''Okpu'', ''nkpara'', ''nza'' or ''odu enyi'' and the traditional flowing shirt known as Isiagu, &lt;ref&gt;{{Cite book |last=Omolade |first=Ajetunmobi |url=https://books.google.com/books?id=9GISAQAAIAAJ |title=Themes in Social Studies Education and Culture: A book of Readings |year=2000 |location=Nigeria |publication-date=2000 |pages=156-166}}&lt;/ref&gt;Akwete or Akwa ocha.&lt;ref name=&quot;:8&quot; /&gt; Although it is made of various materials, the titled men specifically use raw leather made handfans which is either called Akupe or ''agu''&lt;ref&gt;{{Cite book |last=Ndimele |first=Ozo-mekuri |title=Four Decades in the Study of Languages and Linguistics in Nigeria: A Festschrift for Kay Williamson |year=2003 |publication-date=2003 |pages=435}}&lt;/ref&gt; Other names generally used to refer to the ceremonial fan is ''nkuku'' and ''nzuzu'' especially among the ''Agbalanze'' titled group of Onitsha. The Akupe is designed and decorated in various ways but particularly in a thick and heavy pattern for the titled men. Their titles or names are written or carved at the surface of the handfans while their vehicle plate numbers are also designed as such for identification.&lt;ref name=&quot;:1&quot; /&gt; The fan is not merely acquired by titled men but bestowed on them during the capping ceremony where candidates are installed as Nze no ozo title holders or red cap chiefs. The final stage of the ceremony involves handing the candidate the ''Okpu nze na ozo'' and the ''Ugo'' feather, a fan and a sword among other insignia.&lt;ref name=&quot;:14&quot; /&gt; Akupe also serves as a ritual symbol in the Igbo mmanwu (masquerade) tradition, particularly in the cultural ceremony known as the [[Ijele Masquerade|Ijele]] dance where a significant personality among the dance group is known as Akupe carrier. While he is not a masquerade, he plays the prominent role of leading the ''Ijele'' with it's symbolic powerful Akupe. The disappearance of either the Akupe or it's bearer is believed to place the Ijele at risk. The Akupe bearer determines the movement of the Ijele, which moves or remains stationary according to the bearer's actions&lt;ref&gt;{{Cite journal |last=Ikemerike |first=Ikechukwu John |last2=Efuruhievwe |first2=Margaret Akpevweogene |date=2023 |title=Globalization as a Threat to Cultural Identity: A Case Study of Igba Ijele Dance Group of Awkuzu, Anambra State |url=https://ssjhis.org/wp-content/uploads/2024/02/33.-Globalization-as-a-Threat-to-Cultural-Identity-A-Case-Study-of-Igba-Ijele-Dance-Group-of-Awkuzu-Anambra-State.pdf |journal=South-South Journal of Humanities and International Studies |volume=6 |issue=1 |pages=475-496}}&lt;/ref&gt;

==== Akupe as a ceremonial regalia and fashion among the women ====
[[File:Akupe- locally made hand fan.jpg|thumb|Akupe made of decorative woolen textiles often used by women including brides and maids of honour ]] In Igbo society, both large hand fans and artistic hand-held fans have traditionally formed part of women's ceremonial presentation, particularly during weddings, festivals or public celebrations. Historical accounts of Igbo marriage and courtship customs as early as 1921, describe Igbo brides and her maids of honour carrying large fans during the marriage ceremony, particularly the ''Nkpu'' rite to cool and keep the bride refreshed after dancing during the ceremony&lt;ref name=&quot;:6&quot; /&gt; In Igbo dances and performance traditions, artistic hand-held fans are used for aesthetics and symbolic purposes where they form part of the ceremonial ensemble&lt;ref&gt;{{Cite journal |last=Obijiaku |first=Chidi |date=2023-02-27 |title=HYBRIDITY IN MODERN NIGERIAN MUSIC: THE CASE OF IGBO CHORAL ART MUSIC |url=https://journal.ru.ac.za/index.php/africanmusic/article/view/2456 |journal=African Music : Journal of the International Library of African Music |volume=11 |issue=4 |pages=25–42 |doi=10.21504/amj.v11i4.2456 |issn=2524-2741}}&lt;/ref&gt;

=== Fly-whisks (''Nza'', ''Odu ebule'') ===
[[File:Fly whisk (AM 2015.34.53-1).jpg|thumb|Fly-whisk (''Nza'' or ''Odu ebule'')]] Fly-whisks are known as ''Nza'' or ''odu ebule'' among the Igbo people. It is a traditional ceremonial accessory and symbol of distinction in Igbo society. It forms part of the regalia of titled men, chiefs, elders, and other person's of rank also integrated into the attire which includes the akupe, headgear, staff among others. Historical accounts of title taking ceremonies describe newly initiated titled men receiving ceremonial objects as part of their elevation into positions of honour and responsibility within the community such as the ''Ozo'' title.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Archeological evidence from Igbo-ukwu, dating to about the 9th century, shows the antiquity flywhisks within Igbo ceremonial culture. The excavations that uncovered a royal burial furnished with elaborate regalia and the reconstructions of the chamber depicted the high ranking individual holding a ceremonial fly-whisk among other insignia of authority. These regalia have been interpreted in studies as markets of prestige, rank, leadership, royalty, and ritual symbolisms in early Igboland.&lt;ref name=&quot;:0&quot; /&gt; Beyond functioning as an insignia among titled men, royalty and chiefs, fly-whisks and related descriptions of the regalia has been recorded as forming part of the attire of Igbo dancers and brides during festivities or ceremonies like dance performances and traditional marriage. According to Basden in 1921, ''Nkpu'' brides during their marriage ceremony carried cow's tail mounted on a leather handle, sometimes accompanied by small mirrors placed in specially carved hand held frames, including large fans which all formed part of their attire in the ceremony.&lt;ref name=&quot;:6&quot; /&gt; ''Odu enyi'' is also used in artistic and performance traditions of the Igbi people like the ''nkpokiti'' dance. Studies of Igbo oral poetry includes the fly-whisks among the symbolic paraphernalia carried by performers as emblems of their artistry. The flywhisks were described as part of the ceremonial props employed in dances, processions, and choral performance where they serve both symbolic and aesthetic functions as they express the Igbo culture.&lt;ref name=&quot;:15&quot; /&gt;


== Module 4: Traditional Dress and Ensembles ==

=== Igbo traditional dress and fashion ===
Igbo traditional dress and fashion for men typically comprise of loose cotton shirt or robe over an ankle-length wrapper, or loin cloths fashioned from various local Igbo fabrics like the Isiagu, Akwete or Akwaocha complemented with ''okpu'', ''akupe'', ''odu enyi'', ''mkpara'', ''ofo'' and adornments with ''aka'' (coral beads) worn around the neck and wrists. These regalia items proclaim status within the society. Royal robes, royal headdresses, silver sword were used to describe the attire of Igbo royalty like the Obi of Onitsha by Nzimiro and Henderson while the red cap chiefs dressed in their own special attire&lt;ref&gt;{{Cite book |last=Ebuziem |first=Cajetan E. |title=Doing Ministry in the Igbo Context: Towards an Emerging Model and Method for the Church in Africa- Foreword by Theophilus Okere |date=2011 |publisher=Peter Lang Inc., International Academic Publishers |isbn=978-1-4331-1154-9 |edition= |series=Bible and Theology in Africa |location=New York}}&lt;/ref&gt;&lt;ref name=&quot;:20&quot;&gt;{{Cite book |last=Nnoromele |first=Salome |url=https://archive.org/details/lifeamongibowome0000nnor |title=Life among the Ibo women of Nigeria |publisher=Lucent books, San Diego |year=1998 |location=San Diego |publication-date=1998}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; &lt;ref name=&quot;:16&quot; /&gt; Among the women, the typical tradition of dress consist of a pair of matching wrappers or double wrappers known as ''eregbor'' na ''ntukwasi'', a blouse called ''efe obi'' made of Akwete or George fabrics and Ichafu (also spelt Ichafo and Icafo). In ordinary circumstances according to M.M Green in 1947, the women wore short wrappers folded around the hips and reaching the knee with a headcloth while for ceremonial functions a blouse or tunic was worn together with the waist wrappers and festive headcloth &lt;ref name=&quot;:21&quot;&gt;{{Cite book |last=Lamb Holmes |first=Venice Judy |title=Nigerian Weaving |publisher=The Roxford Press |year=1981 |pages=247-280}}&lt;/ref&gt;￼&lt;ref name=&quot;:22&quot;&gt;{{Cite journal |last=Kent |first=Kate P. |last2=Eicher |first2=Joanne Bubolz |last3=Dendel |first3=Esther Warner |date=1978 |title=Nigerian Handcrafted Textiles |url=https://doi.org/10.2307/3335408 |journal=African Arts |volume=11 |issue=3 |pages=14 |doi=10.2307/3335408 |issn=0001-9933}}&lt;/ref&gt;&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; Igbo women complement their dressing with ''aka'' (beads) and Jewellries especially for festivities or ceremonial occasions like traditional marriage known as ''Igba nkwu''.The beadded accessories include ''aka'' or ''nkalari'' (coral beads) worn around the neck and wrists and used as earrings, ''mgbaji'' (flat circular waist beads) worn around the waist, and ''ola'' (iron bangles) around the ankles.&lt;ref name=&quot;:19&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt;

==== Layered Identical Double Wrappers (''Eregbor na Ntukwasi''), Blouse (''Efe obi'') and Ichafu ====
Traditionally, Igbo women's textiles like Akwete of all category are woven in pairs of identical design as well as sold in pairs. These pairs are not sewn together but worn together. The two pairs of wrappers are known as ''eregbor na ntukwasi,'' made of multicolored geometric or floral design. The first wrapper is wrapped around the waist and extends down to the ankle. The second wrapper overlaps the first from the waist to the knees, giving the wrappers a layered look. The wrappers are paired with a blouse known as ''efe obi'' which s tucked inside. This combination of wrappers and blouse are paired with head ties described by Chimamanda Adichie as Ichafu. The textiles used are usually Akwete and George.&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:21&quot; /&gt;&lt;ref name=&quot;:22&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; In recounting her past, Chimamanda Ngozi Adichie describes how she had seen her mother dress up in her double wrappers, blouse and Ichafu. ￼ 
{{Blockquote|text=She folded and twisted and pinned her Ichafu until it sat on her head like a large flower. She wrapped her George - heavy beaded cloth, alive with embroidery, always in bright shades of red or purple pink - around her waist in two layers, The first, the longer piece, hit her ankles, and the second formed an elegant tier just below her knees. Her sequinned blouse caught the light and glittered. Her shoes and handbag always matched|author=Chimamanda Ngozi Adichie|title=Style and Substance: Why What We Wear Matters}} 
This attire ensemble of layered double wrappers called ''eregbor na'' ''ntukwasi'' paired with a blouse and Ichafu (Ichafo) head tie is also complemented with jewelleries as described by Chimamanda and Uzo Aduba who had witnessed their mother and aunt dress up.&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:18&quot; /&gt; [[File:Igbo woman wearing Jooji obiakwa(double wrapper) with uweobi and Ichafu.jpg|left|thumb|A full clothing ensemble of the Igbo woman featuring ''Eregbor na ntukwasi'' (Layered identical George wrappers) paired with a puffed sleeved blouse ''(efe elu)'' and an Ichafu (Ichafo) headdress]] [[File:An igbo woman in Ichafu headdress.jpg|thumb|An Igbo woman wearing a fitted blouse (''efe obi'')]]


== See also ==
* [[Igbo culture]]
== ==

[[Category:Igbo culture]]
[[Category:Material Culture]]
[[Category:Learning Modules]]</text>
      <sha1>ef4dqz0szsxy897gte2ulj9w4xdr6jz</sha1>
    </revision>
    <revision>
      <id>2834716</id>
      <parentid>2834714</parentid>
      <timestamp>2026-09-27T20:54:38Z</timestamp>
      <contributor>
        <username>Joshey11</username>
        <id>3111726</id>
      </contributor>
      <comment>/* Layered Identical Double Wrappers (Eregbor na Ntukwasi), Blouse (Efe obi) and Ichafu */</comment>
      <origin>2834716</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="34714" sha1="nieci7k1rymwtooprd39pxjgu0abdp3" xml:space="preserve">{{Wikipedia|w:Igbo regalia and Headdresses}}

{{course}}
== Introduction ==
'''Igbo regalia and headdresses''' comprise the ceremonial headgear, hairstyles, adornments, accessories, clothing and insignia traditionally associated with the Igbo people. These include mainly the ichafu (head ties), aka (beads), helmet and okpu (caps), Other significant headdresses and regalia include: akupe (hand fans), nza (flywhisks), ofo (elephant tusks), nkpara (staffs ), ugo (eagle's feathers), wigs and other symbolic objects used to express cultural identity, social status, title, spiritual significance and ceremonial functions.&lt;ref name=&quot;:0&quot;&gt;{{Cite book |last=Shaw |first=Thurstan |title=Unearthing Igbo-Ukwu: archaeological discoveries in eastern Nigeria |date=1977 |publisher=Oxford University Press |isbn=978-0-19-575251-9 |location=Ibadan, Nigeria ; New York}}&lt;/ref&gt;&lt;ref name=&quot;:11&quot;&gt;{{Cite book |last=Church Missionary Society |url=https://babel.hathitrust.org/cgi/pt?id=ien.35556041684267&amp;seq=160 |title=Dictionary of the Ibo Language: English-Ibo |publisher=Church Missionary Society Bookshop |year=1923 |location=Lagos |pages=160}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal |last=Dike |first=P. Chike |date=1987 |title=Art, Symbol and Authority Among the Aro of South-east Nigeria |journal=Nigerian Magazine |volume=55 |pages=30-35}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot;&gt;{{Cite book |last=Poynor |first=Robin |url=http://archive.org/details/africanartatharn0000poyn |title=African art at the Harn Museum : spirit eyes, human hands |date=1995 |publisher=Gainesville : University of Florida |others=Internet Archive |isbn=978-0-8130-1325-1 |pages=115}}&lt;/ref&gt; Many forms of Igbo regalia are associated with rulers, chiefs, and titled individuals. They are used to mark rank, status, office and authority, and remain an important part of [[Igbo culture|Igbo cultural]] traditions.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Archeological excavations at Igbo-Ukwu by Charles Thurstan Shaw uncovered 9th-century burials containing ceremonial regalia, including beads, Headdresses, crowns, fly-whisks, anklets, and other prestige objects. &lt;ref name=&quot;:0&quot; /&gt;These discoveries show the long-standing use of adornments and regalia among the Igbo people.

== Learning Objectives ==
Before studying this, you should be able to:
* Differentiate between the cultural roles, historical origins, and gender associations of various Igbo headdresses and items of regalia.
* Understand the socio-political significance of specific insignia like the Okpu ozo, Okpu agu, and Akupe.
* Describe the material culture, artistic elements, and components of traditional Igbo clothing ensembles for both men and women.


== Module 1: Women's Headdresses, Coiffures, and Crowns ==

=== Gẹ̀lẹ̀ headdress ===
[[File:Beauty Queen Bianca.jpg|alt=Bianca Ojukwu wearing Ichafu|thumb|Bianca Odumegwu Ojukwu, an Igbo woman and Nigerian Minister of Foreign Affairs wearing '''GẸ̀LẸ̀''' headdress.]] [[File:Igbo woman styled in Igbo Ichafu (headscarf) and Akwete obiakwa. Stunning.jpg|thumb|Igbo woman wearing '''GẸ̀LẸ̀''' headdress boldly and stylishly tied with a damask head-tie fabric]] [[File:Eze Obi Ossai and wives wearing Ichafu with wrappers and elephant tusk Ivory (Odu).jpg|alt=An image of Igbo women in 1841 wearing Ichafu headdress, Odu ukwu, and wrappers and carrying Akupe while the Obi carries mkpara muo.|thumb|Igbo women described by [[William Allen (Royal Navy officer)|William Allen]] in his 1841 book as Eze Obi's wives. The image depicts them wearing headcloths known as Ichafu among the Igbo, and elephant ivory anklets known as ''odu''. They carry ''Akupe'' (handfans) while the Obi carries a staff known as ''nkpara mmuo'']] '''GẸ̀LẸ̀''' of the Yoruba people of South Western Nigeria. The GẸ̀LẸ̀ as seen above is a more elaborate mode of tying a headtie usually more voluminous than the ichafu. The GẸ̀LẸ̀ requires atleast two yards of fabric to tie on the head unlike the ichafu that requires one yard of fabric to tie on the head.

'''GẸ̀LẸ̀''' be tied with the Ankara, Aso-ofi, adire, damask fabrics, etc. In the mid 90's, the '''GẸ̀LẸ̀''' design was introduced to the Igbos through movement from the south eastern Nigeria to the Western Nigeria. Yoruba fashion was considered trending fashion which eventually have ride to Igbos adopting the '''GẸ̀LẸ̀''' 

'''Ichafu''' (also recorded as Ichafo and Icafo in historical sources) is a traditional Igbo women's headdress fashioned from fabric folded simply and not elaborate sitting on the head. It is a prominent part of ceremonial dresses typically worn by igbo women with traditional outfits. Chimamanda Ngozi Adichie in a piece republished in Style and Substance: Why What We Wear Matters, edited by Gay Garnet narrated watching her mother arrange gèlè on her head until it sat on her head like a large flower&lt;ref name=&quot;:17&quot;&gt;{{Cite book |title=Style and substance: why what we wear matters |date=2023 |publisher=John Murray |isbn=978-1-3998-1246-7 |editor-last=Garnett |editor-first=Bay |location=London}}&lt;/ref&gt; &lt;ref&gt;{{Cite book |last=Butticci |first=Annalisa |title=African Pentecostals in Catholic Europe: the politics of presence in the twenty-first century |date=2016 |publisher=Harvard University Press |isbn=978-0-674-73709-9 |location=Cambridge, Massachusetts}}&lt;/ref&gt;Likewise, Uzo Aduba in her memoir The road is Good described wearing an elaborate gèlè on her head in the most gorgeous coral colour.&lt;ref name=&quot;:18&quot;&gt;{{Cite book |last=Aduba |first=Uzo |title=The road is good: how a mother's strength became a daughter's purpose |date=2024 |publisher=Viking |isbn=978-0-593-29912-8 |location=New York}}&lt;/ref&gt; '''Gèlè''' has influenced contemporay fashion design beyond [[Nigeria]]. A review in the [[fashion]] section of the British newspaper The Observer of Priya Ahluwalia's Spring/Summer 2026 collection noted that the designer's Jacquard knits drew inspiration from Nigerian  headwraps alongside Bollywood motifs&lt;ref&gt;{{Cite web |last=Deaman |first=Jo Jones, Helen Seamons, Sam |title=Hometown glory: highlights from London Fashion Week sprin... |url=https://observer.co.uk/style/fashion/article/hometown-glory-highlights-from-london-fashion-week-springsummer-2026 |access-date=2026-06-20 |website=The Observer |language=en}}&lt;/ref&gt; '''Gèlè''' is also included in an Oxford University Press International Baccalaureate Visual Arts curriculum under the theme &quot;Textiles and cultural signs&quot; alongside Japanese kimonos.&lt;ref&gt;{{Cite book |last=Oxford |url=http://archive.org/details/visual-arts-paterson-poppy-and-vaughn-oxford-2017 |title=Visual Arts Paterson, Poppy And Vaughn Oxford 2017 Textbook |date=2017}}&lt;/ref&gt; Headcloths formed an important part of traditional Igbo women's headdresses and attire. Historical and linguistic sources record terms such as Ichafu, Icafo, Ichafo, Ichafu isi in reference to headcloths, head ties, headgears and headdresses worn by Igbo women in both everyday and ceremonial settings. This headdress was associated with markets, meetings, religious observances, weddings, celebrations, and other social occasions.&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:12&quot;&gt;{{Cite book |last=Cowen |first=Rhoda |url=https://archive.org/details/goldsilverthread0000cowe |title=The Gold and Silver Threads: A memoir of Life in the Twentieth Century |publisher=Alan Sutton Publishing Ltd |year=1994 |location=Stroud, Gloucestershire, England |publication-date=1994 |pages=67, 84}}&lt;/ref&gt;&lt;ref name=&quot;:13&quot;&gt;{{Cite book |last=Green |first=M M |url=https://archive.org/detailsibovillageaffair0000gree/page/218/mode/1up |title=Ibo Village Affairs |date=1947 |publisher=Sidgwick and Jackson |year=1947 |location=Sidgwick London |publication-date=1947 |pages=136, 218, 226}}&lt;/ref&gt; Headcloths are part of a long-standing tradition of Igbo women's headdresses and full clothing ensemble. In the ethnographic work of the anthropologist M.M. Green in Igboland, he documented women in 1947 wearing festive headcloths to market and observed that headcloths were commonly worn at meetings and social gatherings as part of their clothing ensemble.&lt;ref name=&quot;:13&quot; /&gt; Igbo headtie is worn in a simple way usually with one yard fabrics. Writing about her experiences in Igboland, Rhoda Cowen described Igbo headtie referred to as Ichafo and Icafo as a simple brightly colored headdress worn in a simple and stylish manner noting that some were wrapped around the head with projecting ends, while others formed striking elements of women's attire alongside vibrant colored fabrics or textiles and gold jewellery.&lt;ref name=&quot;:12&quot; /&gt; Igbo headtie has also been recorded in ethnographic studies of Igbo dress as a headgear fashioned from a decorative piece of cloth worn around the head known as Ichafo. It is fashioned from a piece of cloth approximately six by three feet in size, folded into triangular or rectangular forms and wrapped around the head in an elevated manner that creates a style resembling a tall turban with decorative bow. According to the author, a headgear is distinguished from a head-tie which is smaller and worn flat around the head in the Nigerian context while defining Igbo style of headtie as a headgear.&lt;ref name=&quot;:9&quot; /&gt;

[[File:Girl on ichafu.jpg|alt=A girl wearing Ịchafú n'isi|thumb|An Igbo girl wearing '''GẸ̀LẸ̀''' headdress]]


=== Helmet coiffures and beaded crowns ===
Igbo headdresses also include elaborate female head adornments, as well as decorative headgear featuring elaborate coiffures worn by masquerades such as the ''Agbogho Mmuo'' (maiden spirit) during festivals and cultural events. The Headdresses and costumes are intended to depict female figures and their feminine appearance and attributes.&lt;ref&gt;{{Cite book |last=Celenko |first=Theodore |title=A treasury of African art from the Harrison Eiteljorg Collection |last2=Eiteljorg |first2=Harrison |date=1983 |publisher=Indiana University Press |isbn=978-0-253-11057-2 |location=Bloomington}}&lt;/ref&gt; [[File:Ancient Igbo helmet Coiffures and headgear (1921).jpg|alt=Helmet Coiffures and Headgears of Ancient Igbo brides|thumb|Ethnographic photos of Igbo brides known as Nkpu brides of prospective chiefs dressed in helmet Coiffures and heagears, necklaces of Leapard teeth and aggry beads 1920.]] Helmet-shaped coiffures were among the elaborate hairstyles historically worn by Igbo women during courtship, marriage festivities, and other ceremonial occasions. Some were built on a foundation of clay, charcoal and palm oil and moulded into a crest resembling the central ridge of a Roman helmet, extending from the forehead to the nape of the neck. The coiffures were often further decorated with beads, small could plaits, cowry shells, mother-of-pearl, brass ornaments, and mirrors sewn into the hair.&lt;ref name=&quot;:6&quot;&gt;{{Cite book |last=Basden |first=George Thomas |url=https://doi.org/10.5479/sil.115290.39088000476515 |title=Among the Ibos of Nigeria |date=1921 |publisher=Seeley, Service &amp; Co., ltd}}&lt;/ref&gt;&lt;ref name=&quot;:7&quot;&gt;{{Cite journal |last=Chudi-Duru |first=Chika C. |date=2024 |title=MMA NWANYI BU EKIKE |url=https://www.journals.ezenwaohaetorc.org/index.php/UJOCC/article/viewFile/3950/4060 |journal=Ohazurume: Unizik Journal of Culture and Civilization (often abbreviated UJOCC) |volume=3 |pages=96-115}}&lt;/ref&gt; Other recorded styles include a raised helmet-like ridge formed on a clay foundation and decorated with beads, cowry shells, leopard claws, camwood paste, and other adornments. Such coiffures could signify age, status, wealth or other stages of life and formed part of ceremonial female adornment in parts of Igboland.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt; [[File:An Igbo bride adorn with the isi agu and the red bead.jpg|alt=An attire featuring Igbo beaded head crown called ngala, aka, and nza|thumb|An Igbo bride adorned with beaded crown called ''ngala'', aka(coral beads) worn around the neck and wrist and worn as earrings and carrying the ''nza'' (flywhisks) regalia.]] While elaborate Helmet-shaped coiffures were a prominent part of historical Igbo women's ceremonial adornment, contemporary ceremonial attire incorporates beaded crowns and bead-based headpieces. Studies of present-day Igbo dress culture depict the brides and her maidens wearing beaded crowns as part of the traditional attire, or decorate their hair with bead accessories during weddings and Cultural celebrations. Beaded crowns, together with corals and other ornamental beads remain a prominent feature of Igbo regalia and are associated with beauty, femininity, cultural identity, fertility, spiritual well-being and marital blessings.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;


== Module 2: Elite Signifiers, Beads, and Men's Okpu ==

=== Aka ===
[[File:Igbo Bride during her traditional marriage 20220216.jpg|alt=Igbo bride adorned with beaded accessories and Uli body arts and carrying a Calabash.|thumb|An Igbo bride dressed for ''Igbankwu'' (traditional wedding) adorned in beaded accessories featuring aka attached on the head, ''nkalari'' or ''erulu'' (coral beads) around the neck, wrists ankles and also worn as earrings. She carries the ''nza'' (fly whisk) regalia and a Calabash with her legs designed in Uli body arts.]] Beaded accessories made of glass beads are known as ''aka'' among the Igbo people. They're of various types which consist of coral beads known as ''erulu'' or ''aka'' and waist beads known as ''mgbaji''. The large coral beads are known as ''nkalari''. The ''mgbaji'' is usually a kind of flat circular coral beads worn around the waist. The large coral beads are worn around the neck, and also worn as earrings and on the wrists.&lt;ref name=&quot;:9&quot;&gt;{{Cite book |last=Melie |first=Edith E. |url=https://books.google.com/books?id=iqZbAAAAMAAJ |title=The Ozo Title of Onitsha: A Study of it's Dress and Insignia |publisher=University of Wisconsin-Madison |year=1977 |location=Madison, Wisconsin, USA |publication-date=1977 |pages=87-140}}&lt;/ref&gt;&lt;ref name=&quot;:19&quot;&gt;{{Cite book |url=https://books.google.com/books?id=UAgOAQAAMAAJ |title=Ikenga |date=1985 |publisher=Institute of African Studies, University of Nigeria. |language=en}}&lt;/ref&gt; Aka is an important part of both Igbo men's and Women's dress fashion. Among the men, it is also regarded as a status symbol across [[Africa]] and a ceremonial adornment like bridal attire for Igbo women. Gold beads are also incorporated in the dress attire.&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt; [[File:Glass beads from Igbo-Ukwu.jpg|thumb|9th Century carbon-dated Igbo-ukwu glass beads.]] Beads are highly valued in Igboland&lt;ref&gt;{{Cite book |last=Afigbo |first=A. E. |title=The Igbo and their neighbours: inter-group relations in southeastern Nigeria to 1953 |date=1987 |publisher=University Press |isbn=978-0-19-575713-2 |location=Ibadan}}&lt;/ref&gt; and have long been an important part of Igbo ceremonial dress and adornment. Excavations at Igbo-ukwu uncovered large quantities of glass and carnelian beads used in necklaces, armlets, wristlets, girdles, and other ornaments. In one royal burial, hundreds of beads were found around the skull, which suggests that the deceased wore a beaded headdress, while strings of beads and a [[copper]] crown was part of the ceremonial regalia. These discoveries dated as early as the 9th century by Thurstan Shaw show that beaded accessories was widely used as a regalia symbol and adornment in ancient Igbo society.&lt;ref name=&quot;:0&quot; /&gt;

=== Okpu ===
Okpu refers to traditional caps or helmets used by Igbo adult males of various statuses and ranks in the society for symbolic purposes. There are different kinds of Okpu worn by adult males in Igboland. The most prominent are feathered red cap known as ''okpu ozo'' and leopard cap known as ''Okpu agu''.&lt;ref name=&quot;:2&quot;&gt;{{Cite book |last=Oriji |first=John Nwachimereze |url=http://archive.org/details/ngwahistorystudy0055orij |title=Ngwa history : a study of social and economic changes in Igbo mini-states in time perspective |date=1991 |publisher=New York : P. Lang |others=Internet Archive |isbn=978-0-8204-1411-9}}&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;{{Cite book |last=M. Angulu Onwuejeogwu |url=http://archive.org/details/an-igbo-civilization-nri-kingdom-and-hegemony |title=An Igbo Civilization: Nri Kingdom and Hegemony |date=1980}}&lt;/ref&gt;

==== Okpu ozo ====
[[File:Red Cap Chiefs at an Igbo Traditional Ceremony.jpg|alt=Ndi Nze na ozo you Okpu ozo ma jidekwa Akupe, ofo, na mkpara|thumb|Titled Igbo men known as Ndi Nze na ozo wearing their traditional regalia featuring Okpu ozo (red caps made of hide with eagle's feathers). Their attire also features other Igbo regalia insignia such as Akupe, ''ofo'', ''nza'', ''nkpara'']] Okpu ozo is described as the feathered red cap worn by titled men known as Ndi Nze na Ozo. As a paraphernalia of office, okpu is regarded as sacred thereby prohibiting ordinary people from touching them.&lt;ref&gt;{{Cite book |last=Oriji |first=J. |url=https://books.google.com/books?id=WZliAQAAQBAJ |title=Political Organization in Nigeria since the Late Stone Age: A History of the Igbo People |date=2011-01-17 |publisher=Springer |isbn=978-0-230-11668-9 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:10&quot;&gt;{{Cite book |url=https://books.google.com/books?id=e-Y6dSeFt_sC |title=Ikenga |date=1980 |publisher=Institute of African Studies, University of Nigeria. |language=en}}&lt;/ref&gt;It is a high-crowned red cap made of hide and usually encircled with eight eagle feather plumes called ''ugo''.This is particularly worn by the title holders called ''Nze''. It can also be modest or low-crowned without the feather decorations worn by the ''Ozo'' title holders.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal |last=Ubani |first=Kenneth |date=2019 |title=Igbo Leadership Through the Visual Arts: Back to the Future |url= |journal=Canadian Social Science |volume=15 |issue=7}}&lt;/ref&gt; It is also known as Okpu ''mmee mmee'' which literally translates to colour of the cap and okpu ''mmee'' which signifies true loyalty. Some Igbo regions also refer to it as ''okpu eze''. &lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Ifemesia |first=C. C. |url=http://archive.org/details/traditionalhuman00ifem |title=Traditional humane living among the Igbo : an historical perspecitve |date=1979 |publisher=Enugu, Nigeria : Fourth Dimension Publishers |others=Internet Archive |isbn=978-978-156-062-0}}&lt;/ref&gt;&lt;ref name=&quot;:3&quot; /&gt; Okpu ozo is regarded as an insignia that protects the wearer when he was away from his lineage or village as well as other travellers.&lt;ref name=&quot;:4&quot; /&gt; The wearers are known as Ndi nze na ozo or red cap chiefs. To obtain the title, a candidate traditionally applied to the red cap chiefs, who supervised the initiation ceremony. During the capping ceremony, the initiate was presented with a red cap and feather as a symbol of his new status and a symbol of authority to enable him perform his duties. This capping stage is also known as ''ikube-okpu''. He also receives other insignia.&lt;ref name=&quot;:14&quot;&gt;{{Cite book |last=Umeasiegbu |first=Rems N. |title=The way we lived: Ibo customs and stories |date=1981 |publisher=Heinemann |isbn=978-0-435-90061-8 |edition=Repr |series=African writers series |location=London}}&lt;/ref&gt;

==== Okpu agu ====
[[File:Igbo kwenu.jpg|alt=Nwoke yi ekike Okpu agu na agba egwu Ohafia|thumb|Ohafia war dance performer wearing Okpu agu]] [[File:Ohafia Igbo Dance Performance Chicago.jpg|alt=Ndi yi Okpu agu agba egwu Ohafia na Chicago|thumb|Ohafia Igbo dance performance in Chicago [[United States]] featuring the Okpu agu Igbo regalia]] Some descriptions of Igbo headdresses highlight that three kinds of helmets were worn: thick ones made of coco-yam stalks, or of the bark of the Achi tree, and fine looking but thin ones made of young palm leaves or raffia hats called ''okpu uturu.''&lt;ref name=&quot;:5&quot;&gt;{{Cite book |last=Talbot |first=Percy Amaury |title=The People's of Southern Nigeria: A sketch of their History, Ethnology and Languages, with an abstract of 1921 Census. |publisher=Oxford University Press, H Milford. |year=1926 |location=London |publication-date=1926 |pages=413, 839}}&lt;/ref&gt; Among these headgears was the okpu agu, a cap associated with Ohafia warrior traditions, known as the leopard cap of bravery. It is round slanting cap made of a knitted wool of black, white and red stripes with a pattern that resemble leopard markings from which the cap derives its name.&lt;ref&gt;{{Cite book |last=McCall |first=John C |url=https://books.google.com/books?id=204eAQAAMAAJ |title=The Ohafia War Dance as Lived Experience |publisher=University of Michigan |year=1992 |location=Ann Arbor, Michigan |pages=37}}&lt;/ref&gt; The okpu agu was an important symbol of warrior achievement in Ohafia. According to historical accounts, the red colour of the cap was traditionally reserved for warriors who had taken heads in battle, or returned with the slain body of a strong animal like the leopard and the caps were dyed with the blood of war victims￼, while the black and white stripe evoked the leopard and it's qualities of strength, agility and martial prowess that were admired in accomplished warriors, as well as the leopard body movement that characterizes the Ohafia War Dance movement.&lt;ref&gt;{{Cite book |last=MacCall |first=John Christensen |title=Dancing histories: heuristic ethnography with the Ohafia Igbo |date=2000 |publisher=University of Michigan Press |isbn=978-0-472-11070-4 |location=Ann Arbor}}&lt;/ref&gt; Okpu agu was manufactured locally by process of bending over, tying and sowing. It is also known as ''Okpu-Aji'' by Nkanu Igbo people and ''Okpu Ojji'' by Abajah and ''Okpu oggu'' (fighting war caps)&lt;ref name=&quot;:5&quot; /&gt;￼


== Module 3: Ceremonial Utensils, Insignia, and Fly-Whisks ==

=== Akupe (hand fan) ===
[[File:Akupe handfan.webp|thumb|Igbo man carrying an Akupe with ofo symbol carved on it. The image also features him wearing ''Okpu'' ''Ozo'' and Isiagu flowing shirt and ''aka'' (coral beads) worn around his wrist.]] Akupe is a ceremonial traditional handfan of the Igbo people which forms part of the clothing ensemble of Igbo men and women. However, it isn't merely used as a [[fashion]] complement but functions as both a practical object and insignia or symbol of status within Igbo society. It is mainly made of raw leather but was also made of other materials like straw, palm products and copper historically. As a paraphernalia of office, it is mainly associated with the ''Nze na ozo'' title holders.&lt;ref name=&quot;:15&quot;&gt;{{Cite book |last=Uzochukwu |first=Sam |title=Traditional funeral poetry of the Igbo |date=2001 |publisher=Lagos University Press |isbn=978-978-017-624-2 |location=Lagos, Nigeria}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The use of akupe as a ceremonial regalia in Igbo land dates back to Igbo-ukwu archeology where copper hand fan with a handle made of wooden shaft along with other status symbols were unearthed from a royal burial chamber of whom was described by Thurstan Shaw as a high ranking titled man or royalty. These excavations which included the fan was dated 9th century&lt;ref name=&quot;:0&quot; /&gt;

==== Akupe as a ceremonial regalia and Fashion among the men ====
The Akupe serves as both a ceremonial insignia and an element of elite male in Igbo Society. It is part of the regalia of titled men like the Ozo title holders, chiefs. and elders. Typically carried at the hand, it complements other symbols of status such as the ''Okpu'', ''nkpara'', ''nza'' or ''odu enyi'' and the traditional flowing shirt known as Isiagu, &lt;ref&gt;{{Cite book |last=Omolade |first=Ajetunmobi |url=https://books.google.com/books?id=9GISAQAAIAAJ |title=Themes in Social Studies Education and Culture: A book of Readings |year=2000 |location=Nigeria |publication-date=2000 |pages=156-166}}&lt;/ref&gt;Akwete or Akwa ocha.&lt;ref name=&quot;:8&quot; /&gt; Although it is made of various materials, the titled men specifically use raw leather made handfans which is either called Akupe or ''agu''&lt;ref&gt;{{Cite book |last=Ndimele |first=Ozo-mekuri |title=Four Decades in the Study of Languages and Linguistics in Nigeria: A Festschrift for Kay Williamson |year=2003 |publication-date=2003 |pages=435}}&lt;/ref&gt; Other names generally used to refer to the ceremonial fan is ''nkuku'' and ''nzuzu'' especially among the ''Agbalanze'' titled group of Onitsha. The Akupe is designed and decorated in various ways but particularly in a thick and heavy pattern for the titled men. Their titles or names are written or carved at the surface of the handfans while their vehicle plate numbers are also designed as such for identification.&lt;ref name=&quot;:1&quot; /&gt; The fan is not merely acquired by titled men but bestowed on them during the capping ceremony where candidates are installed as Nze no ozo title holders or red cap chiefs. The final stage of the ceremony involves handing the candidate the ''Okpu nze na ozo'' and the ''Ugo'' feather, a fan and a sword among other insignia.&lt;ref name=&quot;:14&quot; /&gt; Akupe also serves as a ritual symbol in the Igbo mmanwu (masquerade) tradition, particularly in the cultural ceremony known as the [[Ijele Masquerade|Ijele]] dance where a significant personality among the dance group is known as Akupe carrier. While he is not a masquerade, he plays the prominent role of leading the ''Ijele'' with it's symbolic powerful Akupe. The disappearance of either the Akupe or it's bearer is believed to place the Ijele at risk. The Akupe bearer determines the movement of the Ijele, which moves or remains stationary according to the bearer's actions&lt;ref&gt;{{Cite journal |last=Ikemerike |first=Ikechukwu John |last2=Efuruhievwe |first2=Margaret Akpevweogene |date=2023 |title=Globalization as a Threat to Cultural Identity: A Case Study of Igba Ijele Dance Group of Awkuzu, Anambra State |url=https://ssjhis.org/wp-content/uploads/2024/02/33.-Globalization-as-a-Threat-to-Cultural-Identity-A-Case-Study-of-Igba-Ijele-Dance-Group-of-Awkuzu-Anambra-State.pdf |journal=South-South Journal of Humanities and International Studies |volume=6 |issue=1 |pages=475-496}}&lt;/ref&gt;

==== Akupe as a ceremonial regalia and fashion among the women ====
[[File:Akupe- locally made hand fan.jpg|thumb|Akupe made of decorative woolen textiles often used by women including brides and maids of honour ]] In Igbo society, both large hand fans and artistic hand-held fans have traditionally formed part of women's ceremonial presentation, particularly during weddings, festivals or public celebrations. Historical accounts of Igbo marriage and courtship customs as early as 1921, describe Igbo brides and her maids of honour carrying large fans during the marriage ceremony, particularly the ''Nkpu'' rite to cool and keep the bride refreshed after dancing during the ceremony&lt;ref name=&quot;:6&quot; /&gt; In Igbo dances and performance traditions, artistic hand-held fans are used for aesthetics and symbolic purposes where they form part of the ceremonial ensemble&lt;ref&gt;{{Cite journal |last=Obijiaku |first=Chidi |date=2023-02-27 |title=HYBRIDITY IN MODERN NIGERIAN MUSIC: THE CASE OF IGBO CHORAL ART MUSIC |url=https://journal.ru.ac.za/index.php/africanmusic/article/view/2456 |journal=African Music : Journal of the International Library of African Music |volume=11 |issue=4 |pages=25–42 |doi=10.21504/amj.v11i4.2456 |issn=2524-2741}}&lt;/ref&gt;

=== Fly-whisks (''Nza'', ''Odu ebule'') ===
[[File:Fly whisk (AM 2015.34.53-1).jpg|thumb|Fly-whisk (''Nza'' or ''Odu ebule'')]] Fly-whisks are known as ''Nza'' or ''odu ebule'' among the Igbo people. It is a traditional ceremonial accessory and symbol of distinction in Igbo society. It forms part of the regalia of titled men, chiefs, elders, and other person's of rank also integrated into the attire which includes the akupe, headgear, staff among others. Historical accounts of title taking ceremonies describe newly initiated titled men receiving ceremonial objects as part of their elevation into positions of honour and responsibility within the community such as the ''Ozo'' title.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Archeological evidence from Igbo-ukwu, dating to about the 9th century, shows the antiquity flywhisks within Igbo ceremonial culture. The excavations that uncovered a royal burial furnished with elaborate regalia and the reconstructions of the chamber depicted the high ranking individual holding a ceremonial fly-whisk among other insignia of authority. These regalia have been interpreted in studies as markets of prestige, rank, leadership, royalty, and ritual symbolisms in early Igboland.&lt;ref name=&quot;:0&quot; /&gt; Beyond functioning as an insignia among titled men, royalty and chiefs, fly-whisks and related descriptions of the regalia has been recorded as forming part of the attire of Igbo dancers and brides during festivities or ceremonies like dance performances and traditional marriage. According to Basden in 1921, ''Nkpu'' brides during their marriage ceremony carried cow's tail mounted on a leather handle, sometimes accompanied by small mirrors placed in specially carved hand held frames, including large fans which all formed part of their attire in the ceremony.&lt;ref name=&quot;:6&quot; /&gt; ''Odu enyi'' is also used in artistic and performance traditions of the Igbi people like the ''nkpokiti'' dance. Studies of Igbo oral poetry includes the fly-whisks among the symbolic paraphernalia carried by performers as emblems of their artistry. The flywhisks were described as part of the ceremonial props employed in dances, processions, and choral performance where they serve both symbolic and aesthetic functions as they express the Igbo culture.&lt;ref name=&quot;:15&quot; /&gt;


== Module 4: Traditional Dress and Ensembles ==

=== Igbo traditional dress and fashion ===
Igbo traditional dress and fashion for men typically comprise of loose cotton shirt or robe over an ankle-length wrapper, or loin cloths fashioned from various local Igbo fabrics like the Isiagu, Akwete or Akwaocha complemented with ''okpu'', ''akupe'', ''odu enyi'', ''mkpara'', ''ofo'' and adornments with ''aka'' (coral beads) worn around the neck and wrists. These regalia items proclaim status within the society. Royal robes, royal headdresses, silver sword were used to describe the attire of Igbo royalty like the Obi of Onitsha by Nzimiro and Henderson while the red cap chiefs dressed in their own special attire&lt;ref&gt;{{Cite book |last=Ebuziem |first=Cajetan E. |title=Doing Ministry in the Igbo Context: Towards an Emerging Model and Method for the Church in Africa- Foreword by Theophilus Okere |date=2011 |publisher=Peter Lang Inc., International Academic Publishers |isbn=978-1-4331-1154-9 |edition= |series=Bible and Theology in Africa |location=New York}}&lt;/ref&gt;&lt;ref name=&quot;:20&quot;&gt;{{Cite book |last=Nnoromele |first=Salome |url=https://archive.org/details/lifeamongibowome0000nnor |title=Life among the Ibo women of Nigeria |publisher=Lucent books, San Diego |year=1998 |location=San Diego |publication-date=1998}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; &lt;ref name=&quot;:16&quot; /&gt; Among the women, the typical tradition of dress consist of a pair of matching wrappers or double wrappers known as ''eregbor'' na ''ntukwasi'', a blouse called ''efe obi'' made of Akwete or George fabrics and Ichafu (also spelt Ichafo and Icafo). In ordinary circumstances according to M.M Green in 1947, the women wore short wrappers folded around the hips and reaching the knee with a headcloth while for ceremonial functions a blouse or tunic was worn together with the waist wrappers and festive headcloth &lt;ref name=&quot;:21&quot;&gt;{{Cite book |last=Lamb Holmes |first=Venice Judy |title=Nigerian Weaving |publisher=The Roxford Press |year=1981 |pages=247-280}}&lt;/ref&gt;￼&lt;ref name=&quot;:22&quot;&gt;{{Cite journal |last=Kent |first=Kate P. |last2=Eicher |first2=Joanne Bubolz |last3=Dendel |first3=Esther Warner |date=1978 |title=Nigerian Handcrafted Textiles |url=https://doi.org/10.2307/3335408 |journal=African Arts |volume=11 |issue=3 |pages=14 |doi=10.2307/3335408 |issn=0001-9933}}&lt;/ref&gt;&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; Igbo women complement their dressing with ''aka'' (beads) and Jewellries especially for festivities or ceremonial occasions like traditional marriage known as ''Igba nkwu''.The beadded accessories include ''aka'' or ''nkalari'' (coral beads) worn around the neck and wrists and used as earrings, ''mgbaji'' (flat circular waist beads) worn around the waist, and ''ola'' (iron bangles) around the ankles.&lt;ref name=&quot;:19&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt;

==== Layered Identical Double Wrappers (''Eregbor na Ntukwasi''), Blouse (''Efe obi'') and Ichafu ====
Traditionally, Igbo women's textiles like Akwete of all category are woven in pairs of identical design as well as sold in pairs. These pairs are not sewn together but worn together. The two pairs of wrappers are known as ''eregbor na ntukwasi,'' made of multicolored geometric or floral design. The first wrapper is wrapped around the waist and extends down to the ankle. The second wrapper overlaps the first from the waist to the knees, giving the wrappers a layered look. The wrappers are paired with a blouse known as ''efe obi'' which s tucked inside. This combination of wrappers and blouse are paired with head ties described by Chimamanda Adichie as Ichafu. The textiles used are usually Akwete and George.&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:21&quot; /&gt;&lt;ref name=&quot;:22&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; In recounting her past, Chimamanda Ngozi Adichie describes how she had seen her mother dress up in her double wrappers, blouse and Ichafu. ￼ 
{{Blockquote|text=She folded and twisted and pinned her Ichafu until it sat on her head like a large flower. She wrapped her George - heavy beaded cloth, alive with embroidery, always in bright shades of red or purple pink - around her waist in two layers, The first, the longer piece, hit her ankles, and the second formed an elegant tier just below her knees. Her sequinned blouse caught the light and glittered. Her shoes and handbag always matched|author=Chimamanda Ngozi Adichie|title=Style and Substance: Why What We Wear Matters}} 
This attire ensemble of layered double wrappers called ''eregbor na'' ''ntukwasi'' paired with a blouse and Ichafu (Ichafo) head tie is also complemented with jewelleries as described by Chimamanda and Uzo Aduba who had witnessed their mother and aunt dress up.&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:18&quot; /&gt; [[File:Igbo woman wearing Jooji obiakwa(double wrapper) with uweobi and Ichafu.jpg|left|thumb|A full clothing ensemble of the Igbo woman featuring ''Eregbor na ntukwasi'' (Layered identical George wrappers) paired with a puffed sleeved blouse ''(efe elu)'' and an Gele (borrowed from Yoruba) headdress]] [[File:An igbo woman in Ichafu headdress.jpg|thumb|An Igbo woman wearing a fitted blouse (''efe obi'')]]


== See also ==
* [[Igbo culture]]
== ==

[[Category:Igbo culture]]
[[Category:Material Culture]]
[[Category:Learning Modules]]</text>
      <sha1>nieci7k1rymwtooprd39pxjgu0abdp3</sha1>
    </revision>
  </page>
  <page>
    <title>The John Snow Prediabetes Institute</title>
    <ns>0</ns>
    <id>330494</id>
    <revision>
      <id>2834828</id>
      <parentid>2834412</parentid>
      <timestamp>2026-09-28T09:41:43Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834828</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6059" sha1="awapl3zzahv6bbtpm2twjo474rlc2qa" xml:space="preserve">The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7).

'''Background:''' 
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). &lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt; All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt; and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs &lt;ref&gt; «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.&lt;/ref&gt; Educational materials come from the international diabetes organisations e.g the ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|&lt;big&gt;
Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt; ]]

===== &lt;big&gt;'''Prevalence studies'''  &lt;/big&gt; =====
[[/The Maritime Health database design // |1.1 The Maritime Health database design]] 

1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

1.5  Excel recoding (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;

'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies'''  Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)  

&lt;big&gt;&lt;br /&gt;
'''Intervention studies''' English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;
&lt;/big&gt;

'''Organisation''' 

- [[/The John Snow Institute bylaws /]] 

- [[/|Manila conference Oct. 2026]] 

- General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;

'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

'''The John Snow Prediabetes Institute Research Network:''' 
&lt;small&gt;Prof. Eng. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain,  Prof. Dr. Olaf Jensen, MD, MPH, PhD;  Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:&lt;/small&gt;


&lt;small&gt;Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark&lt;/small&gt;

==References==

&lt;references /&gt;
Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
      <sha1>awapl3zzahv6bbtpm2twjo474rlc2qa</sha1>
    </revision>
    <revision>
      <id>2834829</id>
      <parentid>2834828</parentid>
      <timestamp>2026-09-28T09:43:18Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834829</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6057" sha1="4jhn2ej16etmnhsf0jlp3gxg3sj7k74" xml:space="preserve">The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7).

'''Background:''' 
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). &lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt; All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt; and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs &lt;ref&gt; «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.&lt;/ref&gt; Educational materials come from the international diabetes organisations e.g the ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|&lt;big&gt;
Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt; ]]

===== &lt;big&gt;'''Prevalence studies'''  &lt;/big&gt; =====
[[/The Maritime Health database design // |1.1 The Maritime Health database design]] 

1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

1.5  Excel recoding (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;

'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies'''  Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)  

&lt;big&gt;&lt;br /&gt;
'''Intervention studies''' English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;
&lt;/big&gt;

'''Organisation''' 

- [[/The John Snow Institute bylaws /]] 

- [[/|Manila conference Oct. 2026]] 

- General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;

'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

'''The John Snow Prediabetes Institute Research Network:''' 
&lt;small&gt;Prof. Eng. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain,  Prof. Dr. Olaf Jensen, MD, MPH, PhD;  Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:&lt;/small&gt; 
&lt;small&gt;Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark&lt;/small&gt;
==References==

&lt;references /&gt;
Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
      <sha1>4jhn2ej16etmnhsf0jlp3gxg3sj7k74</sha1>
    </revision>
    <revision>
      <id>2834831</id>
      <parentid>2834829</parentid>
      <timestamp>2026-09-28T09:58:12Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834831</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6077" sha1="orvmy56b7ls9dy6d4d2mm8l5h0ls9x7" xml:space="preserve">The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7).

'''Background:''' 
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). &lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt; All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt; and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs &lt;ref&gt; «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.&lt;/ref&gt; Educational materials come from the international diabetes organisations e.g the ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|&lt;big&gt;
Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt; ]]

===== &lt;big&gt;'''Prevalence studies'''  &lt;/big&gt; =====
[[/The Maritime Health database design // |1.1 The Maritime Health database design]] 

1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

1.5  Excel recoding (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;

'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies'''  Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)  

&lt;big&gt;&lt;br /&gt;
'''Intervention studies''' English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;
&lt;/big&gt;

'''Organisation''' 

- [[/The John Snow Institute bylaws /]] 

- [[/|Manila conference Oct. 2026]] 

- General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;

'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

'''The John Snow Prediabetes Institute Research Network:''' 
&lt;small&gt;Prof. Eng. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain,  Prof. Dr. Olaf Jensen, MD, MPH, PhD;  Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:&lt;/small&gt; 
&lt;small&gt;Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching&lt;/small&gt; 
==References==

&lt;references /&gt;
Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
      <sha1>orvmy56b7ls9dy6d4d2mm8l5h0ls9x7</sha1>
    </revision>
    <revision>
      <id>2834841</id>
      <parentid>2834831</parentid>
      <timestamp>2026-09-28T11:43:26Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834841</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6126" sha1="thtb5hd03f7ums0wihfalzr6b1bnyee" xml:space="preserve">The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7).

'''Background:''' 
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). &lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt; All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt; and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs &lt;ref&gt; «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.&lt;/ref&gt; Educational materials come from the international diabetes organisations e.g the ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|&lt;big&gt;
Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt; ]]

===== &lt;big&gt;'''Prevalence studies'''  &lt;/big&gt; =====
[[/The Maritime Health database design // |1.1 The Maritime Health database design]] 

1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

1.5  Excel recoding (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;

'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies'''  Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)  

&lt;big&gt;&lt;br /&gt;
'''Intervention studies''' English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;
&lt;/big&gt;

'''Organisation''' 

- [[/The John Snow Institute bylaws /]] 

- [[/|Manila conference Oct. 2026]] 

- General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;

'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

'''The John Snow Prediabetes Institute Research Network:''' 
&lt;small&gt;Prof. Eng. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain,  Prof. Dr. Olaf Jensen, MD, MPH, PhD;  Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:&lt;/small&gt; 
&lt;small&gt;Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;&lt;/small&gt; 
==References==

&lt;references /&gt;
Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
      <sha1>thtb5hd03f7ums0wihfalzr6b1bnyee</sha1>
    </revision>
    <revision>
      <id>2834842</id>
      <parentid>2834841</parentid>
      <timestamp>2026-09-28T11:45:30Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834842</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6126" sha1="dv5tdoro25eerddfvlpudj7x4t1n7d1" xml:space="preserve">The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7).

'''Background:''' 
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). &lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt; All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer &lt;ref&gt;https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 &lt;/ref&gt; and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs &lt;ref&gt; «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.&lt;/ref&gt; Educational materials come from the international diabetes organisations e.g the ADA:&lt;ref&gt;https://professional.diabetes.org/diabetes-support-resources&lt;/ref&gt; and the Health Promoting Schools.&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;[[File:Lifestyle Medicine Pillars.png|250px|right|&lt;big&gt;
Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars&lt;/big&gt; ]]

===== &lt;big&gt;'''Prevalence studies'''  &lt;/big&gt; =====
[[/The Maritime Health database design // |1.1 The Maritime Health database design]] 

1.2  Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.4. School childrens` Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

1.5  Excel recoding (pre diabetes, hypertension etc) &lt;ref&gt;https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG&lt;/ref&gt;

'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies'''  Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)  

&lt;big&gt;&lt;br /&gt;
'''Intervention studies''' English 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;
&lt;/big&gt;

'''Organisation''' 

- [[/The John Snow Institute bylaws /]] 

- [[/|Manila conference Oct. 2026]] 

- General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;

'''Publications'''
&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

'''The John Snow Prediabetes Institute Research Network:''' 
&lt;small&gt;Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain,  Prof. Dr. Olaf Jensen, MD, MPH, PhD;  Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:&lt;/small&gt; 
&lt;small&gt;Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;&lt;/small&gt; 
==References==

&lt;references /&gt;
Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
      <sha1>dv5tdoro25eerddfvlpudj7x4t1n7d1</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Impulsivity versus sensation-seeking</title>
    <ns>0</ns>
    <id>331004</id>
    <revision>
      <id>2834696</id>
      <parentid>2834441</parentid>
      <timestamp>2026-09-27T18:53:24Z</timestamp>
      <contributor>
        <username>Jaspershields</username>
        <id>3005613</id>
      </contributor>
      <minor />
      <comment>/* Understanding impulsivity */</comment>
      <origin>2834696</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="29314" sha1="34wzc6v4w3mgqremgwpkn59z824krzs" xml:space="preserve">{{title|Impulsivity versus sensation-seeking:&lt;br&gt;What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour?}}
__TOC__

{{RoundBoxTop|theme=2}}
[[File: Beer Cantus.jpg|right|thumb|250px|'''Figure 1.''' Social drinking can involve similar observable behaviours despite differences in the psychological processes underlying an individual's decision to participate.]]

'''Imagine this...'''

At a university party, two students are offered an unfamiliar alcoholic beverage. The first accepts immediately, without hesitation, considering the possible consequences. The second pauses but deliberately chooses to try it because the experience is new and exciting. From the outside, their behaviour looks almost identical: both accept the drink. Psychologically, however, the processes motivating their decisions may be quite different.
{{RoundBoxBottom}}

Impulsivity and sensation seeking are related but distinct psychological constructs that can help explain why people engage in seemingly similar behaviours for different reasons. Impulsivity is multidimensional and includes tendencies such as acting without sufficient forethought, whereas sensation seeking reflects the pursuit of novel, exciting, or stimulating experiences (Goh et al., 2020; Ravert &amp; Donnellan, 2021). Consequently, risk taking behaviour does not necessarily indicate that an individual is simply &quot;impulsive&quot;. Understanding the motivation and psychological processes behind the behaviour can provide a more complete explanation of why it occurs.

Distinguishing impulsivity from sensation seeking is particularly essential when considering risk taking behaviour. Different dimensions of impulsivity and sensation seeking are associated with different patterns of health risk behaviour, suggesting that similar actions may arise through varied psychological pathways (Rogers et al., 2021). In the opening scenario, for instance, immediately accepting the drink without considering its consequences may reflect a lack of premeditation, whereas deliberately accepting it for novelty and excitement may more closely reflect sensation seeking. The behavioural outcome is similar, but the motivation is not.

Alcohol use provides a useful real world example of why this distinction matters. Research with university students suggests that sensation seeking and impulsivity can relate to alcohol involvement through different pathways, with sensation seeking more strongly associated with alcohol consumption and impulsivity more closely associated with alcohol related problems (Magid et al., 2007). Identifying these differences can therefore improve understanding not only of whether someone engages in risky behaviour but also of the psychological processes that may contribute to that behaviour and its consequences.

This chapter examines the theoretical and empirical distinction between impulsivity and sensation seeking and evaluates how each may influence behaviour. By examining their conceptual overlap, underlying psychological processes, relationships with risk taking, and associations with alcohol use, the chapter demonstrates that understanding ''why'' a person behaves in a particular way can be as important as observing ''what'' they do.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is impulsivity and what are its key characteristics?
* What is sensation seeking and how does it differ from impulsivity?
* What psychological processes distinguish impulsivity from sensation seeking?
* How do impulsivity and sensation seeking differentially influence risk-taking behaviour?
* How can these differences help explain alcohol use and related consequences?
{{RoundBoxBottom}}

== Understanding impulsivity ==

Impulsivity is commonly understood as acting without thinking, but psychological research suggests that it is not a single characteristic. Behaviours described as impulsive can result from varied psychological processes, such as acting impetuously during intense emotions, failing to consider future consequences, or struggling to persist with difficult tasks. Research supports this multidimensional understanding, finding that measures labelled as impulsivity assess several related but distinguishable characteristics rather than one unified trait (Berg et al., 2015; Sharma et al., 2014). Thus, understanding why an individual acts impulsively may provide more information than simply labelling the behaviour as impulsive.

=== The UPPS-P model ===

One influential framework for understanding these differences is the UPPS-P model of impulsive personality. The original UPPS model proposed four dimensions: negative urgency, or acting rashly during negative emotion; lack of premeditation, or acting without adequately considering consequences, lack of perseverance, involving difficulty remaining focused on demanding or boring tasks and sensation seeking, involving preference for exciting and novel experiences (Whiteside &amp; Lynam, 2001). The framework was later expanded to distinguish positive urgency, referring to hastely actions during intense positive emotion, from negative urgency (Cyders &amp; Smith, 2008). Consequently, two apparently impulsive decisions may originate from quite different combinations of emotion, planning, persistence and reward seeking.

Experts agree that these components should be measured individually. Network analysis of the UPPS-P found meaningful relationships between its components while also indicating that the dimensions are not interchangeable (Goh et al., 2020). Other research has continued to examine and refine multidimensional measures of impulsivity, demonstrating both the usefulness and continuing complexity of defining the construct (Samiefard et al., 2023; Türkmen et al., 2023). This is an important limitation of the broad term impulsivity: although multidimensional models provide greater precision, disagreement about exactly how impulsivity should be organised and measured remains.

=== Emotion and urgency ===

[[File:Positive &amp; Negative Urgency.png|right|thumb|400px|'''Figure 2'''. Positive and negative urgency describe tendencies towards rash action during intense positive and negative emotional states. Although the emotional states differ, both illustrate how intense emotion may contribute to impulsive action (Cyders &amp; Smith, 2008).]]

The distinction between positive and negative urgency further demonstrates why motivation matters. Urgency theory proposes that intense emotional states can increase the likelihood of reckless action, with positive urgency occurring during strongly positive emotion and negative urgency during strongly negative emotion (Cyders &amp; Smith, 2008). This differs from lack of premeditation because a person may ordinarily consider consequences effectively but behave differently when experiencing intense emotion.

The opening party scenario illustrates this distinction. Immediately accepting the unfamiliar drink without adequately considering its consequences may indicate lack of premeditation. If the decision instead occurred during intense excitement at the party, positive urgency might provide another explanation. The observable action alone therefore cannot identify which dimension was responsible.

&lt;quiz display=simple&gt;
{According to the UPPS-P model, impulsivity is best understood as a single characteristic that affects behaviour in the same way across different situations.}
- True
+ False

{Positive urgency refers to rash action during intense positive emotion, whereas negative urgency refers to rash action during intense negative emotion.}
+ True
- False
&lt;/quiz&gt;

== Understanding sensation seeking ==

Sensation seeking is strongly correlated to impulsivity but places greater emphasis on the motivation to experience novelty, intensity and stimulation. The construct has traditionally been understood as an individual difference in preference for varied, novel, complex and intense experiences, sometimes accompanied by willingness to accept risk to obtain those experiences (Roberti, 2004; Zuckerman &amp; Kuhlman, 2000). Ultimately indicating that risk itself does not always have to be the person's goal. Instead, risk may be accepted because it accompanies an experience considered sufficiently interesting or rewarding.

=== A motivational perspective ===

This provides an important distinction from other impulsivity dimensions. Someone displaying lack of premeditation may act before adequately considering the consequences, whereas a sensation seeker may recognise the consequences and deliberately decide that the potential stimulation is worthwhile. In the opening scenario, the second student hesitates before accepting the unfamiliar drink but chooses it because the experience is new and exciting. The decision may therefore involve forethought while still being associated with risk.

Sensation seeking should consequently not be treated as inherently dysfunctional. Ravert and Donnellan (2021) found different associations between impulsivity, sensation seeking and psychological well being, supporting the idea that the constructs should not simply be combined into a general tendency towards problematic behaviour. Sensation seeking may encourage exploration and engagement as well as potentially hazardous behaviour, meaning that its consequences depend partly on how and where the need for stimulation is expressed.

=== A theoretical complication ===

An important theoretical complication is that sensation seeking is simultaneously treated as a distinct construct and included as a dimension within the UPPS-P model of impulsivity (Whiteside &amp; Lynam, 2001). The inclusion of sensation seeking within the UPPS-P model does not mean that it is the same as other forms of impulsivity; rather, it is considered one distinct pathway that may contribute to impulsive behaviour. Contrastingly, it demonstrates that the boundaries surrounding the broader concept of impulsivity remain debated and depend partly on how researchers define and measure it. If all sensation seeking were simply impulsivity, there would be little reason to expect the two to show different relationships with behaviour. Evidence that they predict different outcomes can therefore help determine whether distinguishing them is psychologically meaningful.

&lt;quiz display=simple&gt;
{Which statement best distinguishes sensation seeking from lack of premeditation?}
- Sensation seeking always involves acting without considering the consequences.
+ Sensation seeking may involve recognising potential consequences but deciding that the novelty or stimulation is worthwhile.
- Lack of premeditation involves deliberately pursuing novel and exciting experiences.
- Sensation seeking and lack of premeditation describe the same psychological process.

{Why is sensation seeking theoretically complicated within models of impulsivity?}
- Sensation seeking has no relationship with impulsive behaviour.
- Sensation seeking has been removed from multidimensional models of impulsivity.
+ Sensation seeking can be treated as a distinct construct while also being included as a dimension within the UPPS-P model.
- Sensation seeking only occurs when an individual does not recognise the risks of their behaviour.
&lt;/quiz&gt;

== Distinguishing impulsivity from sensation seeking ==

[[File:Sensation seeking Vs Impulsivity.png|right|thumb|387px|'''Figure 3'''. Comparison of the UPPS-P dimensions of impulsive behaviour and Zuckerman’s dimensions of sensation seeking, highlighting areas of conceptual overlap and distinction (Goh et al., 2020; Magid et al., 2007; Samiefard et al., 2023). (image created independently using Canva software).]]

Impulsivity and sensation seeking overlap because both can contribute to behaviours involving uncertainty or risk. The key distinction concerns the process motivating the action. Dimensions such as urgency and lack of premeditation emphasise rash action or insufficient regulation, whereas sensation seeking emphasises approach towards novelty, excitement and stimulation (Cyders &amp; Smith, 2008; Zuckerman &amp; Kuhlman, 2000). Similar behaviour can therefore emerge through different psychological pathways.

Research comparing the constructs provides support for this distinction. Magid et al. (2007), found that sensation seeking and impulsivity showed different relationships with alcohol involvement, suggesting that combining them could overlook meaningful differences. Similarly, a meta-analysis found that sensation seeking and impulsivity were both associated with greater risk taking, but sensation seeking showed a somewhat stronger relationship with risky behaviour than impulsivity (Lauriola et al., 2014). These findings support distinguishing the constructs while also demonstrating that neither provides a complete explanation of risky behaviour.

'''Table 1''' 

'''Table – Comparing impulsivity and sensation''' '''seeking''' 

The comparison table summarises the major conceptual differences and similarities between impulsivity and sensation seeking, including their defining characteristics, underlying processes, and relationships with risk taking behaviour.
{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! Feature !! Impulsivity !! Sensation seeking
|-
| '''Core characteristic''' || Tendency toward rash or insufficiently considered action across multiple dimensions || Tendency to seek novel, varied, exciting or stimulating experiences
|-
| '''Key Process''' || Can involve reduced forethought, urgency or behavioural regulation || Motivation toward novelty, excitement and stimulation
|-
|'''Risk Taking'''
|Risk may result from insufficient consideration of consequences or other impulsive tendencies
|Risk may be accepted in pursuit of rewarding or stimulating experiences
|-
|'''UPPS-P'''
|Multidimensional model including positive urgency, negative urgency, lack of premeditation, lack of perseverance and sensation seeking
|Sensation seeking is one distinct dimension within the UPPS-P model
|-
|'''Example'''
|Immediately accepting an unfamiliar drink without considering consequences
|Deliberately trying the drink because the experience is novel and exciting
|-
|'''Behavioural Outcome'''
|Can contribute to health risk, gambling and substance related behaviours
|Can contribute to risk taking, alcohol use and novelty oriented behaviour
|}
Whilst impulsivity and sensation seeking can contribute to similar observable behaviours, their underlying characteristics and psychological processes differ (see Table 1; Goh et al., 2020; Magid et al., 2007).

=== Development and the dual systems model ===

The dual systems model provides another theoretical perspective for understanding this distinction. The model proposes that heightened reward sensitivity develops differently from capacities associated with cognitive control, potentially contributing to increased risk taking during adolescence (Steinberg, 2010). From this perspective, attraction towards rewarding or stimulating experiences and the ability to regulate behaviour are related but separable processes.

Conversely, the model has also been critically reconsidered. Shulman et al. (2016) concluded that evidence broadly supports different developmental trajectories for reward related and cognitive control systems while also highlighting complexities that cannot be captured by a simple imbalance between two systems. Risk taking varies substantially across people and situations, so developmental models should not be interpreted as suggesting that risky behaviour is inevitable or caused by one neural mechanism. The theory is useful for explaining why reward seeking and behavioural control can be distinguished, but it should complement rather than replace personality and contextual explanations.

== Impulsivity, sensation seeking, and risk taking behaviour ==

[[File:Boxing in Uruguay - Palacio Peñarol.jpg|right|thumb|350px|'''Figure 4'''. Activities involving risk and stimulation, such as boxing, illustrate that engagement in risky behaviour does not necessarily indicate impulsivity. The psychological motivation underlying the behaviour may be important for distinguishing impulsivity from sensation seeking.]]

The theoretical distinction becomes particularly useful when considering risk taking behaviour. If impulsivity and sensation seeking represented the same underlying characteristic, they would be expected to show broadly equivalent relationships with risky outcomes. Instead, evidence indicates that different dimensions can be associated with different behaviours and consequences.

Research examining risk taking has found that both sensation seeking and impulsivity are associated with a greater tendency to engage in risky behaviour, although these relationships are relatively modest (Lauriola et al., 2014). This pattern extends across varied types of behaviour, with sensation seeking and impulsivity related traits associated with health risk behaviours (Rogers et al., 2021), adolescent risk taking (Siraj et al., 2021), and speeding behaviour (Sârbescu &amp; Rusu, 2021). More recently, Grubbs et al. (2024) demonstrated that the specific psychological trait involved may also relate to different aspects of risky behaviour. Sensation seeking was associated with whether individuals had engaged in sports wagering, while urgency related dimensions were associated with different patterns of wagering involvement and gambling related problems. 

These findings provide evidence for the theoretical distinction, whilst also requiring cautious interpretation. Personality traits generally explain only part of the variation in complex behaviours, and correlations cannot establish that a trait directly causes an individual to take a particular risk. Social influences, opportunity, emotional state, perceived rewards and the specific type of risk may interact with personality characteristics. Therefore, describing someone simply as a &quot;risk taker&quot; may obscure both the psychological pathway and the context contributing to their behaviour.

== Alcohol use and related consequences ==

Alcohol use provides an especially useful real-world example because drinking and experiencing alcohol related problems are not the same outcome. Sensation seeking may motivate approach towards stimulating social or drinking experiences, whereas other impulsivity dimensions may be more relevant to rash decisions and harmful consequences. This distinction directly reflects the opening scenario: both students accept the same drink, however their reasons for doing so differ.

Magid et al. (2007) found that sensation seeking and impulsivity related differently to alcohol use and alcohol related problems among university students. Broader research evidence supports this outcome specific approach; for instance across 96 studies, Coskunpinar et al. (2013) found that different UPPS dimensions showed different relationships with alcohol outcomes, with urgency dimensions particularly relevant to alcohol related problems. Evidence from individuals entering residential alcohol detoxification also demonstrates clinically relevant relationships between UPPS-P dimensions and substance related characteristics (Kempeneers et al., 2023).

These findings illustrate why psychological explanations should expand beyond asking whether someone is simply &quot;impulsive&quot;. Identifying whether behaviour is motivated primarily by stimulation, emotional urgency, insufficient forethought or another process may provide a more precise understanding of the behaviour. Simultaniously, much of this evidence is correlational and relies on self report measures, so personality should not be interpreted as determining whether an individual will drink or experience alcohol related harm.

=== From explanation to application ===

Understanding different pathways may also have practical implications. Individuals motivated primarily by sensation seeking may benefit from finding safer ways to obtain novelty and stimulation, whereas behaviour associated with urgency may require greater attention to managing intense emotional states before acting. Lack of premeditation is particularly relevant, strategies that introduce time for considering consequences may be more appropriate.

These possibilities illustrate why distinguishing psychological processes can contribute to improving everyday behaviour rather than merely attaching personality labels to people. However, individualised strategies should not be inferred solely from personality scores, and the available evidence does not demonstrate that matching interventions to UPPS-P dimensions will necessarily produce better outcomes. Instead, the distinction provides a framework for understanding why the same behaviour may require different explanations and potentially different approaches to change.


'''Summary Quiz: Impulsivity or sensation seeking?'''
&lt;quiz display=&quot;simple&quot;&gt;
{A student is offered an unfamiliar alcoholic drink at a party and immediately accepts without considering the possible consequences. Which characteristic best explains this behaviour?
|type=&quot;()&quot;}

+ Lack of premeditation
- Thrill and adventure seeking
- Experience seeking
- Boredom susceptibility

{A student deliberately chooses to try an unfamiliar activity because they enjoy novelty, excitement, and stimulation. Which construct best explains this motivation?
|type=&quot;()&quot;}

- Lack of perseverance
+ Sensation seeking
- Negative urgency
- Lack of premeditation

{Which statement best describes the distinction between impulsivity and sensation seeking?
|type=&quot;()&quot;}

- Impulsivity and sensation seeking are interchangeable terms for the same psychological construct.
- Sensation seeking always involves acting without considering consequences.
+ Similar observable behaviours can arise from different underlying motivations and psychological processes.
- Only impulsivity is associated with risk taking behaviour.
&lt;/quiz&gt;

== Conclusion ==
Impulsivity and sensation seeking are related psychological constructs, however should not automatically be treated as interchangeable. Multidimensional models such as the UPPS-P demonstrate that behaviour commonly labelled &quot;impulsive&quot; can arise through different pathways involving emotion, forethought, persistence and attraction towards stimulation. Sensation seeking is particularly important because it overlaps conceptually with impulsivity while emphasising the motivation to pursue novel and exciting experiences.

Research supports the practical value of distinguishing these processes. Sensation seeking and different dimensions of impulsivity show different patterns of association with risk taking, alcohol involvement and other potentially harmful behaviours, although these relationships are generally probabilistic rather than deterministic. Consequently, observable behaviour alone cannot reveal why an individual acted as they did.

The opening scenario illustrates the practical significance of distinguishing between these psychological processes. Accepting an unfamiliar drink may result from acting without considering consequences, seeking an exciting new experience, responding rashly to intense emotion, or a combination of these processes. Understanding why behaviour occurs, rather than simply labelling what occurred, can provide a more useful foundation for understanding and potentially improving everyday decision making.

== See also ==
* [[Motivation and emotion/Book/2011/Sensation seeking|Sensation seeking]] (Book chapter, 2011)
* [[w:Sensation Seeking Scale|Sensation Seeking Scale]] (Wikipedia)

== References ==
{{Hanging indent|1=

Berg, J. M., Latzman, R. D., Bliwise, N. G., &amp; Lilienfeld, S. O. (2015). Parsing the heterogeneity of impulsivity: A meta-analytic review of the behavioral implications of the UPPS for psychopathology. ''Psychological Assessment, 27''(4), 1129–1146. https://doi.org/10.1037/pas0000111

Coskunpinar, A., Dir, A. L., &amp; Cyders, M. A. (2013). Multidimensionality in impulsivity and alcohol use: A meta-analysis using the UPPS model of impulsivity. ''Alcoholism: Clinical and Experimental Research, 37''(9), 1441–1450. https://doi.org/10.1111/acer.12131

Cyders, M. A., &amp; Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin, 134''(6), 807–828. https://doi.org/10.1037/a0013341

Goh, P. K., Lee, C. A., Martel, M. M., Fillmore, M. T., Derefinko, K. J., &amp; Lynam, D. R. (2020). Conceptualizing the UPPS-P model of impulsive personality through network analysis: Key dimensions and general robustness across young adulthood. ''Journal of Personality, 88''(6), 1302–1314. https://doi.org/10.1111/jopy.12572

Grubbs, J. B., Connolly, A. J., Floyd, C. G., Kim, H. S., &amp; Kraus, S. W. (2024). Impulsivity and diverse forms of sports wagering in the U.S.: An examination of the UPPS-P model. ''Addictive Behaviors, 156'', 108058. https://doi.org/10.1016/j.addbeh.2024.108058

Kempeneers, P., Mreyen, K., Pallincourt, R., Remacle, F., Wildemeersch, G., &amp; Simon, J. (2023). Validation of the UPPS-P Impulsive Behavior Scale and clinical correlates of its scores in French-speaking patients starting a residential detoxification program. ''Indian Journal of Psychological Medicine, 45''(5), 503–510. https://doi.org/10.1177/02537176231157411

Lauriola, M., Panno, A., Levin, I. P., &amp; Lejuez, C. W. (2014). Individual differences in risky decision making: A meta-analysis of sensation seeking and impulsivity with the Balloon Analogue Risk Task. ''Journal of Behavioral Decision Making, 27''(1), 20–36. https://doi.org/10.1002/bdm.1784

Magid, V., MacLean, M. G., &amp; Colder, C. R. (2007). Differentiating between sensation seeking and impulsivity through their mediated relations with alcohol use and problems. ''Addictive Behaviors, 32''(10), 2046–2061. https://doi.org/10.1016/j.addbeh.2007.01.015

Ravert, R. D., &amp; Donnellan, M. B. (2021). Impulsivity and sensation seeking: Differing associations with psychological well-being. ''Applied Research in Quality of Life, 16''(4), 1503–1515. https://doi.org/10.1007/s11482-020-09829-y

Roberti, J. W. (2004). A review of behavioral and biological correlates of sensation seeking. ''Journal of Research in Personality, 38''(3), 256–279. https://doi.org/10.1016/S0092-6566(03)00067-9

Rogers, M. M., Kelley, K., &amp; McKinney, C. (2021). Trait impulsivity and health risk behaviors: A latent profile analysis. ''Personality and Individual Differences, 171'', 110511. https://doi.org/10.1016/j.paid.2020.110511

Samiefard, M., Salehi Fadardi, J., Kareshki, H., &amp; Stacy, A. W. (2023). Validity and reliability of a revised S-UPPS-P Impulsive Behavior Scale: The interplay between impulsivity and working memory. ''Journal of Personality Assessment, 105''(2), 174–186. https://doi.org/10.1080/00223891.2022.2081922

Sârbescu, P., &amp; Rusu, A. (2021). Personality predictors of speeding: Anger-aggression and impulsive-sensation seeking. A systematic review and meta-analysis. ''Journal of Safety Research, 77'', 86–98. https://doi.org/10.1016/j.jsr.2021.02.004

Sharma, L., Markon, K. E., &amp; Clark, L. A. (2014). Toward a theory of distinct types of “impulsive” behaviors: A meta-analysis of self-report and behavioral measures. ''Psychological Bulletin, 140''(2), 374–408. https://doi.org/10.1037/a0034418

Shulman, E. P., Smith, A. R., Silva, K., Icenogle, G., Duell, N., Chein, J., &amp; Steinberg, L. (2016). The dual systems model: Review, reappraisal, and reaffirmation. ''Developmental Cognitive Neuroscience, 17'', 103–117. https://doi.org/10.1016/j.dcn.2015.12.010

Siraj, R., Najam, B., &amp; Ghazal, S. (2021). Sensation seeking, peer influence, and risk-taking behavior in adolescents. ''Education Research International, 2021'', 8403024. https://doi.org/10.1155/2021/8403024

Steinberg, L. (2010). A dual systems model of adolescent risk-taking. ''Developmental Psychobiology, 52''(3), 216–224. https://doi.org/10.1002/dev.20445

Türkmen, İ., Rodoplu, N., Üner, B. S., Esmer, Ş. C., Altan-Atalay, A., &amp; Ece, B. (2023). When the UPPS-P model of impulsivity meets a revised approach: The development and validation of the TRUE Multidimensional Impulsivity Scale. ''Journal of Personality Assessment, 105''(3), 355–370. https://doi.org/10.1080/00223891.2022.2093730

Whiteside, S. P., &amp; Lynam, D. R. (2001). The five factor model and impulsivity: Using a structural model of personality to understand impulsivity. ''Personality and Individual Differences, 30''(4), 669–689. https://doi.org/10.1016/S0191-8869(00)00064-7

Zuckerman, M., &amp; Kuhlman, D. M. (2000). Personality and risk-taking: Common biosocial factors. ''Journal of Personality, 68''(6), 999–1029. https://doi.org/10.1111/1467-6494.00124

}}

==External links==
* [https://dictionary.apa.org/sensation-seeking-scale APA Dictionary of Psychology – Sensation-Seeking Scale]
* [https://onlinelibrary.wiley.com/doi/full/10.1002/9781405186407.wbiecs029 Sensation Seeking – Zuckerman] {{ic|Move academic sources to citations/references}}
* [https://www.impulsivity.org/measurement/upps_p/ UPPS-P Impulsive Behavior Scale]

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Impulsivity]]
[[Category:Motivation and emotion/Book/Sensation seeking]]</text>
      <sha1>34wzc6v4w3mgqremgwpkn59z824krzs</sha1>
    </revision>
    <revision>
      <id>2834774</id>
      <parentid>2834696</parentid>
      <timestamp>2026-09-28T01:54:00Z</timestamp>
      <contributor>
        <username>U3279062</username>
        <id>3106301</id>
      </contributor>
      <comment>editing across page</comment>
      <origin>2834774</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="29004" sha1="3d19lt13c5k6g0a8se3erclxt8eaq3t" xml:space="preserve">{{title|Impulsivity versus sensation-seeking:&lt;br&gt;What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour?}}
__TOC__

{{RoundBoxTop|theme=2}}
[[File: Beer Cantus.jpg|right|thumb|250px|'''Figure 1.''' Social drinking can involve similar observable behaviours despite differences in the psychological processes underlying an individual's decision to participate.]]

'''Imagine this...'''

At a university party, two students are offered an unfamiliar alcoholic beverage. The first accepts immediately, without hesitation, considering the possible consequences. The second pauses but deliberately chooses to try it because the experience is new and exciting. From the outside, their behaviour looks almost identical: both accept the drink. Psychologically, however, the processes motivating their decisions may be quite different.
{{RoundBoxBottom}}

Impulsivity and sensation seeking are related but distinct psychological constructs that can help explain why people engage in seemingly similar behaviours for different reasons. Impulsivity is multidimensional and includes tendencies such as acting without sufficient forethought, whereas sensation seeking reflects the pursuit of novel, exciting, or stimulating experiences (Goh et al., 2020; Ravert &amp; Donnellan, 2021). Consequently, risk-taking behaviour does not necessarily indicate that an individual is &quot;impulsive&quot;. Understanding the motivation and psychological processes behind the behaviour can better explain why it occurs.

Distinguishing impulsivity from sensation seeking is particularly important when considering risk-taking behaviour. Different dimensions of impulsivity and sensation seeking are associated with different patterns of health risk behaviour, suggesting that similar actions may arise through varied psychological pathways (Rogers et al., 2021). In the opening scenario, for instance, immediately accepting the drink without considering its consequences may reflect a lack of premeditation, whereas deliberately accepting it for novelty and excitement may more closely reflect sensation seeking. The behavioural outcome is similar, but the motivation is not.

Alcohol use provides a useful real-world example of why this distinction matters. Research with university students suggests that sensation seeking and impulsivity can relate to alcohol involvement through different pathways, with sensation seeking more strongly associated with alcohol consumption and impulsivity more closely associated with alcohol related problems (Magid et al., 2007). Identifying these differences can therefore improve understanding not only of whether someone engages in risky behaviour but also of the psychological processes that may contribute to that behaviour and its consequences.

This chapter examines the theoretical and empirical distinction between impulsivity and sensation seeking and evaluates how each may influence behaviour. By examining their conceptual overlap, underlying psychological processes, links to risk-taking, and associations with alcohol use, the chapter shows that understanding ''why'' a person behaves a certain way can be as important as observing ''what'' they do.{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is impulsivity and what are its key characteristics?
* What is sensation seeking and how does it differ from impulsivity?
* What psychological processes distinguish impulsivity from sensation seeking?
* How do impulsivity and sensation seeking differentially influence risk-taking behaviour?
* How can these differences help explain alcohol use and related consequences?
{{RoundBoxBottom}}

== Understanding impulsivity ==

Impulsivity is commonly understood as acting without thinking, but psychological research suggests that it is not a single characteristic. Behaviours described as impulsive can result from varied psychological processes, such as acting impetuously during intense emotions, failing to consider future consequences, or struggling to persist with difficult tasks. Research supports this multidimensional understanding, finding that measures labelled as impulsivity assess several related but distinguishable characteristics rather than one unified trait (Berg et al., 2015; Sharma et al., 2014). Thus, understanding why an individual acts impulsively may provide more information than simply labelling the behaviour as impulsive.

=== The UPPS-P model ===

One influential framework for understanding these differences is the UPPS-P model of impulsive personality. The original UPPS model proposed four dimensions: negative urgency, or acting rashly during negative emotion; lack of premeditation, or acting without adequately considering consequences; lack of perseverance, involving difficulty remaining focused on demanding or boring tasks; and sensation seeking, involving a preference for exciting and novel experiences (Whiteside &amp; Lynam, 2001). The framework was later expanded to distinguish positive urgency, referring to hasty actions during intense positive emotion, from negative urgency (Cyders &amp; Smith, 2008). Consequently, two apparently impulsive decisions may originate from quite different combinations of emotion, planning, persistence and reward seeking.

Experts agree that these components should be measured individually. Network analysis of the UPPS-P found meaningful relationships among its components and indicated that the dimensions are not interchangeable (Goh et al., 2020). Other research has continued to examine and refine multidimensional measures of impulsivity, demonstrating both the usefulness and continuing complexity of defining the construct (Samiefard et al., 2023; Türkmen et al., 2023). This is an important limitation of the broad term impulsivity: although multidimensional models provide greater precision, disagreement remains about how impulsivity should be organised and measured.

=== Emotion and urgency ===

[[File:Positive &amp; Negative Urgency.png|right|thumb|400px|'''Figure 2'''. Positive and negative urgency describe tendencies towards rash action during intense positive and negative emotional states. Although the emotional states differ, both illustrate how intense emotion may contribute to impulsive action (Cyders &amp; Smith, 2008).]]

The distinction between positive and negative urgency further demonstrates why motivation matters. Urgency theory proposes that intense emotional states can increase the likelihood of reckless action, with positive urgency occurring during strongly positive emotion and negative urgency during strongly negative emotion (Cyders &amp; Smith, 2008). This differs from lack of premeditation because a person may ordinarily consider consequences effectively but behave differently when experiencing intense emotion.

The opening party scenario illustrates this distinction. Immediately accepting the unfamiliar drink without adequately considering its consequences may indicate lack of premeditation. If the decision instead occurred during intense excitement at the party, positive urgency might provide another explanation.  The observable action alone, therefore, cannot identify which dimension was responsible.

&lt;quiz display=simple&gt;
{According to the UPPS-P model, impulsivity is best understood as a single characteristic that affects behaviour in the same way across different situations.}
- True
+ False

{Positive urgency refers to rash action during intense positive emotion, whereas negative urgency refers to rash action during intense negative emotion.}
+ True
- False
&lt;/quiz&gt;

== Understanding sensation seeking ==

Sensation seeking is strongly correlated with impulsivity but emphasises the motivation to experience novelty, intensity, and stimulation. The construct has traditionally been understood as an individual difference in preference for varied, novel, complex, and intense experiences, sometimes accompanied by a willingness to accept risk to obtain them (Roberti, 2004; Zuckerman &amp; Kuhlman, 2000). This ultimately indicates that risk itself does not always have to be the person's goal. Instead, people may accept risk because it accompanies an experience they find sufficiently interesting or rewarding.

=== A motivational perspective ===

This distinguishes it from other dimensions of impulsivity. Someone displaying a lack of premeditation may act before adequately considering the consequences, whereas a sensation seeker may recognise the consequences and deliberately decide that the potential stimulation is worthwhile. In the opening scenario, the second student hesitates before accepting the unfamiliar drink but chooses it because the experience is new and exciting. The decision may therefore involve forethought while still carrying risk.

Sensation seeking should consequently not be treated as inherently dysfunctional. Ravert and Donnellan (2021) found different associations between impulsivity, sensation seeking, and psychological well-being, supporting the idea that the constructs should not simply be combined into a general tendency toward problematic behaviour. Sensation seeking may encourage exploration and engagement as well as potentially hazardous behaviour, meaning that its consequences depend partly on how and where the need for stimulation is expressed.

=== A theoretical complication ===

A key theoretical complication is that sensation seeking is treated as both a distinct construct and a dimension within the UPPS-P model of impulsivity (Whiteside &amp; Lynam, 2001). Including sensation seeking within the UPPS-P model does not mean it is the same as other forms of impulsivity; rather, it is considered a distinct pathway that may contribute to impulsive behaviour. In contrast, it shows that the boundaries around the broader concept of impulsivity remain debated and depend partly on how researchers define and measure it. If all sensation seeking were simply impulsivity, there would be little reason to expect the two to show different relationships with behaviour. Evidence that they predict different outcomes can therefore help determine whether distinguishing them is psychologically meaningful.

&lt;quiz display=simple&gt;
{Which statement best distinguishes sensation seeking from lack of premeditation?}
- Sensation seeking always involves acting without considering the consequences.
+ Sensation seeking may involve recognising potential consequences but deciding that the novelty or stimulation is worthwhile.
- Lack of premeditation involves deliberately pursuing novel and exciting experiences.
- Sensation seeking and lack of premeditation describe the same psychological process.

{Why is sensation seeking theoretically complicated within models of impulsivity?}
- Sensation seeking has no relationship with impulsive behaviour.
- Sensation seeking has been removed from multidimensional models of impulsivity.
+ Sensation seeking can be treated as a distinct construct while also being included as a dimension within the UPPS-P model.
- Sensation seeking only occurs when an individual does not recognise the risks of their behaviour.
&lt;/quiz&gt;

== Distinguishing impulsivity from sensation seeking ==

[[File:Sensation seeking Vs Impulsivity.png|right|thumb|387px|'''Figure 3'''. Comparison of the UPPS-P dimensions of impulsive behaviour and Zuckerman’s dimensions of sensation seeking, highlighting areas of conceptual overlap and distinction (Goh et al., 2020; Magid et al., 2007; Samiefard et al., 2023). (image created independently using Canva software).]]

Impulsivity and sensation seeking overlap because both can contribute to behaviours involving uncertainty or risk. The key distinction concerns the process motivating the action. Dimensions such as urgency and lack of premeditation emphasise rash action or poor regulation, whereas sensation seeking emphasises an approach to novelty, excitement, and stimulation (Cyders &amp; Smith, 2008; Zuckerman &amp; Kuhlman, 2000). Similar behaviour can therefore emerge through different psychological pathways.

Research comparing these constructs supports this distinction. Magid et al. (2007) found that sensation seeking and impulsivity showed different relationships with alcohol involvement, suggesting that combining them could overlook meaningful differences. Similarly, a meta-analysis found that sensation seeking and impulsivity were both associated with greater risk-taking, but sensation seeking showed a somewhat stronger relationship with risky behaviour than impulsivity (Lauriola et al., 2014). These findings support distinguishing the constructs while also showing that neither fully explains risky behaviour.

'''Table 1'''

'''Table – Comparing impulsivity and sensation''' '''seeking''' 

The comparison table summarises the major conceptual differences and similarities between impulsivity and sensation seeking, including their defining characteristics, underlying processes, and relationships with risk taking behaviour.
{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! Feature !! Impulsivity !! Sensation seeking
|-
| '''Core characteristic''' || Tendency toward rash or insufficiently considered action across multiple dimensions || Tendency to seek novel, varied, exciting or stimulating experiences
|-
| '''Key Process''' || Can involve reduced forethought, urgency or behavioural regulation || Motivation toward novelty, excitement and stimulation
|-
|'''Risk Taking'''
|Risk may result from insufficient consideration of consequences or other impulsive tendencies
|Risk may be accepted in pursuit of rewarding or stimulating experiences
|-
|'''UPPS-P'''
|Multidimensional model including positive urgency, negative urgency, lack of premeditation, lack of perseverance and sensation seeking
|Sensation seeking is one distinct dimension within the UPPS-P model
|-
|'''Example'''
|Immediately accepting an unfamiliar drink without considering consequences
|Deliberately trying the drink because the experience is novel and exciting
|-
|'''Behavioural Outcome'''
|Can contribute to health risk, gambling and substance related behaviours
|Can contribute to risk taking, alcohol use and novelty oriented behaviour
|}
Whilst impulsivity and sensation seeking can contribute to similar observable behaviours, their underlying characteristics and psychological processes differ (see Table 1; Goh et al., 2020; Magid et al., 2007).

=== Development and the dual systems model ===

The dual systems model offers another theoretical perspective on this distinction. The model proposes that heightened reward sensitivity develops differently from capacities associated with cognitive control, potentially contributing to increased risk-taking during adolescence (Steinberg, 2010). From this perspective, attraction towards rewarding or stimulating experiences and the ability to regulate behaviour are related but separable processes.

Conversely, the model has also been critically reconsidered. Shulman et al. (2016) concluded that evidence broadly supports different developmental trajectories for reward-related and cognitive control systems, while also highlighting complexities that a simple imbalance between the two systems cannot capture. Risk-taking varies substantially across people and situations, so developmental models should not be interpreted as suggesting that risky behaviour is inevitable or caused by one neural mechanism. The theory helps explain why reward seeking and behavioural control can be distinguished, but it should complement rather than replace personality and contextual explanations.

== Impulsivity, sensation seeking, and risk taking behaviour ==

[[File:Boxing in Uruguay - Palacio Peñarol.jpg|right|thumb|350px|'''Figure 4'''. Activities involving risk and stimulation, such as boxing, illustrate that engagement in risky behaviour does not necessarily indicate impulsivity. The psychological motivation underlying the behaviour may be important for distinguishing impulsivity from sensation seeking.]]

The theoretical distinction is particularly useful when considering risk-taking behaviour. If impulsivity and sensation seeking represented the same underlying characteristic, they would be expected to show broadly equivalent relationships with risky outcomes. Instead, evidence indicates that different dimensions can be associated with different behaviours and consequences.

Research examining risk-taking has found that both sensation seeking and impulsivity are associated with a greater tendency to engage in risky behaviour, although these relationships are relatively modest (Lauriola et al., 2014). This pattern extends across behaviour types, with sensation seeking and impulsivity-related traits associated with health risk behaviours (Rogers et al., 2021), adolescent risk-taking (Siraj et al., 2021), and speeding behaviour (Sârbescu &amp; Rusu, 2021). More recently, Grubbs et al. (2024) demonstrated that the specific psychological trait involved may also relate to different aspects of risky behaviour. Sensation seeking was associated with whether individuals had engaged in sports wagering, while urgency-related dimensions were associated with different patterns of wagering involvement and gambling-related problems.

These findings support the theoretical distinction but require cautious interpretation. Personality traits generally explain only part of the variation in complex behaviours, and correlations cannot establish that a trait directly causes an individual to take a particular risk. Social influences, opportunity, emotional state, perceived rewards and the specific type of risk may interact with personality characteristics. Therefore, describing someone simply as a &quot;risk taker&quot; may obscure both the psychological pathway and the context contributing to their behaviour.

== Alcohol use and related consequences ==

Alcohol use provides an especially useful real-world example because drinking and experiencing alcohol related problems are not the same outcome. Sensation seeking may motivate approach towards stimulating social or drinking experiences, whereas other impulsivity dimensions may be more relevant to rash decisions and harmful consequences. This distinction directly reflects the opening scenario: both students accept the same drink; however, their reasons for doing so differ.

Magid et al. (2007) found that sensation seeking and impulsivity related differently to alcohol use and alcohol related problems among university students. Broader research evidence supports this outcome-specific approach; for instance, across 96 studies, Coskunpinar et al. (2013) found that different UPPS dimensions showed different relationships with alcohol outcomes, with urgency dimensions particularly relevant to alcohol related problems. Evidence from individuals entering residential alcohol detoxification also demonstrates clinically relevant relationships between UPPS-P dimensions and substance-related characteristics (Kempeneers et al., 2023).

These findings illustrate why psychological explanations should expand beyond asking whether someone is &quot;impulsive&quot;. Identifying whether behaviour is motivated primarily by stimulation, emotional urgency, insufficient forethought, or another process may provide a more precise understanding of the behaviour. At the same time, much of this evidence is correlational. It relies on self-report measures, so personality should not be interpreted as determining whether an individual will drink or experience alcohol related harm.

=== From explanation to application ===

Understanding different pathways may also have practical implications. Individuals motivated primarily by sensation seeking may benefit from finding safer ways to obtain novelty and stimulation. In contrast, behaviour associated with urgency may require greater attention to managing intense emotional states before acting. Lack of premeditation is particularly relevant; strategies that build in time to consider consequences may be more appropriate.

These possibilities illustrate why distinguishing psychological processes can improve everyday behaviour rather than merely attaching personality labels to people. However, individualised strategies should not be inferred solely from personality scores, and the available evidence does not demonstrate that matching interventions to UPPS-P dimensions will necessarily produce better outcomes. Instead, the distinction provides a framework for understanding why the same behaviour may require different explanations and potentially different approaches to change.


'''Summary Quiz: Impulsivity or sensation seeking?'''&lt;quiz display=&quot;simple&quot;&gt;
{A student is offered an unfamiliar alcoholic drink at a party and immediately accepts without considering the possible consequences. Which characteristic best explains this behaviour?
|type=&quot;()&quot;}

+ Lack of premeditation
- Thrill and adventure seeking
- Experience seeking
- Boredom susceptibility

{A student deliberately chooses to try an unfamiliar activity because they enjoy novelty, excitement, and stimulation. Which construct best explains this motivation?
|type=&quot;()&quot;}

- Lack of perseverance
+ Sensation seeking
- Negative urgency
- Lack of premeditation

{Which statement best describes the distinction between impulsivity and sensation seeking?
|type=&quot;()&quot;}

- Impulsivity and sensation seeking are interchangeable terms for the same psychological construct.
- Sensation seeking always involves acting without considering consequences.
+ Similar observable behaviours can arise from different underlying motivations and psychological processes.
- Only impulsivity is associated with risk taking behaviour.
&lt;/quiz&gt;

== Conclusion ==
Impulsivity and sensation seeking are related psychological constructs; however, they should not automatically be treated as interchangeable. Multidimensional models such as the UPPS-P show that behaviour commonly labelled &quot;impulsive&quot; can arise through different pathways involving emotion, forethought, persistence, and attraction to stimulation. Sensation seeking is particularly important because it overlaps conceptually with impulsivity while emphasising the motivation to pursue novel and exciting experiences.

Research supports the practical value of distinguishing these processes. Sensation seeking and different dimensions of impulsivity show different patterns of association with risk-taking, alcohol involvement and other potentially harmful behaviours. However, these relationships are generally probabilistic rather than deterministic. Consequently, observable behaviour alone cannot reveal why an individual acted as they did.

The opening scenario illustrates the practical significance of distinguishing between these psychological processes. Accepting an unfamiliar drink may result from acting without considering consequences, seeking an exciting new experience, responding rashly to intense emotion, or a combination of these processes. Understanding why behaviour occurs, rather than simply labelling what occurred, can provide a more useful foundation for understanding and potentially improving everyday decision-making.

== See also ==
* [[Motivation and emotion/Book/2011/Sensation seeking|Sensation seeking]] (Book chapter, 2011)
* [[w:Sensation Seeking Scale|Sensation Seeking Scale]] (Wikipedia)

== References ==
{{Hanging indent|1=

Berg, J. M., Latzman, R. D., Bliwise, N. G., &amp; Lilienfeld, S. O. (2015). Parsing the heterogeneity of impulsivity: A meta-analytic review of the behavioral implications of the UPPS for psychopathology. ''Psychological Assessment, 27''(4), 1129–1146. https://doi.org/10.1037/pas0000111

Coskunpinar, A., Dir, A. L., &amp; Cyders, M. A. (2013). Multidimensionality in impulsivity and alcohol use: A meta-analysis using the UPPS model of impulsivity. ''Alcoholism: Clinical and Experimental Research, 37''(9), 1441–1450. https://doi.org/10.1111/acer.12131

Cyders, M. A., &amp; Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin, 134''(6), 807–828. https://doi.org/10.1037/a0013341

Goh, P. K., Lee, C. A., Martel, M. M., Fillmore, M. T., Derefinko, K. J., &amp; Lynam, D. R. (2020). Conceptualizing the UPPS-P model of impulsive personality through network analysis: Key dimensions and general robustness across young adulthood. ''Journal of Personality, 88''(6), 1302–1314. https://doi.org/10.1111/jopy.12572

Grubbs, J. B., Connolly, A. J., Floyd, C. G., Kim, H. S., &amp; Kraus, S. W. (2024). Impulsivity and diverse forms of sports wagering in the U.S.: An examination of the UPPS-P model. ''Addictive Behaviors, 156'', 108058. https://doi.org/10.1016/j.addbeh.2024.108058

Kempeneers, P., Mreyen, K., Pallincourt, R., Remacle, F., Wildemeersch, G., &amp; Simon, J. (2023). Validation of the UPPS-P Impulsive Behavior Scale and clinical correlates of its scores in French-speaking patients starting a residential detoxification program. ''Indian Journal of Psychological Medicine, 45''(5), 503–510. https://doi.org/10.1177/02537176231157411

Lauriola, M., Panno, A., Levin, I. P., &amp; Lejuez, C. W. (2014). Individual differences in risky decision making: A meta-analysis of sensation seeking and impulsivity with the Balloon Analogue Risk Task. ''Journal of Behavioral Decision Making, 27''(1), 20–36. https://doi.org/10.1002/bdm.1784

Magid, V., MacLean, M. G., &amp; Colder, C. R. (2007). Differentiating between sensation seeking and impulsivity through their mediated relations with alcohol use and problems. ''Addictive Behaviors, 32''(10), 2046–2061. https://doi.org/10.1016/j.addbeh.2007.01.015

Ravert, R. D., &amp; Donnellan, M. B. (2021). Impulsivity and sensation seeking: Differing associations with psychological well-being. ''Applied Research in Quality of Life, 16''(4), 1503–1515. https://doi.org/10.1007/s11482-020-09829-y

Roberti, J. W. (2004). A review of behavioral and biological correlates of sensation seeking. ''Journal of Research in Personality, 38''(3), 256–279. https://doi.org/10.1016/S0092-6566(03)00067-9

Rogers, M. M., Kelley, K., &amp; McKinney, C. (2021). Trait impulsivity and health risk behaviors: A latent profile analysis. ''Personality and Individual Differences, 171'', 110511. https://doi.org/10.1016/j.paid.2020.110511

Samiefard, M., Salehi Fadardi, J., Kareshki, H., &amp; Stacy, A. W. (2023). Validity and reliability of a revised S-UPPS-P Impulsive Behavior Scale: The interplay between impulsivity and working memory. ''Journal of Personality Assessment, 105''(2), 174–186. https://doi.org/10.1080/00223891.2022.2081922

Sârbescu, P., &amp; Rusu, A. (2021). Personality predictors of speeding: Anger-aggression and impulsive-sensation seeking. A systematic review and meta-analysis. ''Journal of Safety Research, 77'', 86–98. https://doi.org/10.1016/j.jsr.2021.02.004

Sharma, L., Markon, K. E., &amp; Clark, L. A. (2014). Toward a theory of distinct types of “impulsive” behaviors: A meta-analysis of self-report and behavioral measures. ''Psychological Bulletin, 140''(2), 374–408. https://doi.org/10.1037/a0034418

Shulman, E. P., Smith, A. R., Silva, K., Icenogle, G., Duell, N., Chein, J., &amp; Steinberg, L. (2016). The dual systems model: Review, reappraisal, and reaffirmation. ''Developmental Cognitive Neuroscience, 17'', 103–117. https://doi.org/10.1016/j.dcn.2015.12.010

Siraj, R., Najam, B., &amp; Ghazal, S. (2021). Sensation seeking, peer influence, and risk-taking behavior in adolescents. ''Education Research International, 2021'', 8403024. https://doi.org/10.1155/2021/8403024

Steinberg, L. (2010). A dual systems model of adolescent risk-taking. ''Developmental Psychobiology, 52''(3), 216–224. https://doi.org/10.1002/dev.20445

Türkmen, İ., Rodoplu, N., Üner, B. S., Esmer, Ş. C., Altan-Atalay, A., &amp; Ece, B. (2023). When the UPPS-P model of impulsivity meets a revised approach: The development and validation of the TRUE Multidimensional Impulsivity Scale. ''Journal of Personality Assessment, 105''(3), 355–370. https://doi.org/10.1080/00223891.2022.2093730

Whiteside, S. P., &amp; Lynam, D. R. (2001). The five factor model and impulsivity: Using a structural model of personality to understand impulsivity. ''Personality and Individual Differences, 30''(4), 669–689. https://doi.org/10.1016/S0191-8869(00)00064-7

Zuckerman, M., &amp; Kuhlman, D. M. (2000). Personality and risk-taking: Common biosocial factors. ''Journal of Personality, 68''(6), 999–1029. https://doi.org/10.1111/1467-6494.00124

}}

==External links==
* [https://dictionary.apa.org/sensation-seeking-scale APA Dictionary of Psychology – Sensation-Seeking Scale]
* [https://onlinelibrary.wiley.com/doi/full/10.1002/9781405186407.wbiecs029 Sensation Seeking – Zuckerman] {{ic|Move academic sources to citations/references}}
* [https://www.impulsivity.org/measurement/upps_p/ UPPS-P Impulsive Behavior Scale]

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Impulsivity]]
[[Category:Motivation and emotion/Book/Sensation seeking]]</text>
      <sha1>3d19lt13c5k6g0a8se3erclxt8eaq3t</sha1>
    </revision>
  </page>
  <page>
    <title>Module:Sandbox/22</title>
    <ns>828</ns>
    <id>331015</id>
    <revision>
      <id>2834741</id>
      <parentid>2834625</parentid>
      <timestamp>2026-09-28T00:13:35Z</timestamp>
      <contributor>
        <username>Helpme2222</username>
        <id>3106525</id>
      </contributor>
      <origin>2834741</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="20852" sha1="sla1koklbc9jehl6uk456n7cxmyesnq" xml:space="preserve">-- My function for teaching grammar

local p = {};
local frame = mw.getCurrentFrame()
local grammatical_moods = {
	&quot;Jelen ido&quot;,
	&quot;Mult ido&quot;,
	&quot;Felszolito mod&quot;,
	&quot;Felteteles mod&quot;
}
local balls = {
	grammatical_moods
}

local toggleButton = mw.html.create( 'span' )
	:css({
		float = &quot;right&quot;,
		[&quot;margin-right&quot;] = &quot;0&quot;,
		[&quot;font-weight&quot;] = &quot;normal&quot;
	})
	:addClass(&quot;mw-customtoggle-Infobox&quot;)
	:node(&quot;[Click to toggle]&quot;)

function p.subheading( discrim, b_content, content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node((discrim) and b_content or content)
			:node((discrim) and toggleButton)
		)
end
	
function p.s_subheading( s_content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible mw-collapsed&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node(s_content)
			:node(toggleButton)
		)
end

function p.row( discrim, name, data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; ) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(name)
		)
		:node( mw.html.create( 'td' )
			:node(data)
		)
end
	
function p.s_row( s_name, s_data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass(&quot;mw-collapsible mw-collapsed&quot;) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(s_name)
		)
		:node( mw.html.create( 'td' )
			:node(s_data)
		)
end

function p.var(symbol)
	return frame:expandTemplate{ title = &quot;Template:Var&quot;, args = {symbol} }
end
	

function p.infoboxWBB()
	-- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html
	--[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node 
			Thankfully, these operations interpret passed nils as no-ops 
			(I'm technically using false though. Somehow that works anyways?)]]
	
	local frame = mw.getCurrentFrame()
	local ballname = frame.args[&quot;Title&quot;] or tostring(mw.title.getCurrentTitle())
	local Entities = p -- require(&quot;Module:Entities&quot;)
	
	local featureCount = tonumber(frame.args[&quot;Feature count&quot;]) or 1
	local s_featureCount = tonumber(frame.args[&quot;SUPER Feature count&quot;]) or featureCount
	local matchCount = {
		total = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local matchWinCount = {
		standard = mw.site.stats.pagesInCategory( ballname..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local s_matchCount = {
		total = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local s_matchWinCount = {
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local isSuper = 
		frame.args[&quot;SUPER Feature count&quot;] or
		frame.args[&quot;SUPER Type&quot;] or 
		frame.args[&quot;SUPER Debuted&quot;] or
		(s_matchCount.total &gt; 0) or 
		nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra &quot;or nil&quot; had to be added since this is used in :addClass calls.
	-- TITLE
	local infobox = mw.html.create( 'table' )
		:addClass(&quot;wikitable&quot;)
		:css({
			float = &quot;right&quot;; 
			[&quot;margin-left&quot;] = &quot;10px&quot;
		})
		:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass((isSuper) and &quot;mw-collapsible&quot; or nil)
				:node(&quot;&lt;big&gt;&quot; .. ballname .. &quot;&lt;/big&gt;&quot;)
			)
		)
		:node((isSuper) and mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass(&quot;mw-collapsible mw-collapsed&quot;)
				:node(&quot;&lt;big&gt;&quot; .. (frame.args[&quot;SUPER Title&quot;] or (&quot;Super &quot; .. ballname)) .. &quot;&lt;/big&gt;&quot;)
			)
		)
	-- SECTION: Identification
		-- Section Heading
		:node( p.subheading( (frame.args[&quot;SUPER Type&quot;]), &quot;Base Identification&quot;, &quot;Basic Info&quot; ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_subheading( &quot;Super Identification&quot; ) )
		-- Type
		:node( p.row( (frame.args[&quot;SUPER Type&quot;]), &quot;Type&quot;, frame.args[&quot;Type&quot;] ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_row( &quot;Type&quot;, frame.args[&quot;SUPER Type&quot;] ) )
		-- Color
	local color = frame.args[&quot;Color&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;color&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Color&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(color ~= &quot;&quot;) and frame:expandTemplate{ title = &quot;Color&quot;, args = {color} } or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add color...&lt;/i&gt;]]&quot;
				)
			)
		)
		-- Emoji
	local emoji = frame.args[&quot;Emoji&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;emoji&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Emoji&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(emoji ~= &quot;&quot;) and ( emoji .. &quot; &lt;code&gt;&quot; .. p.emojiCodepointInternal(emoji) .. &quot;&lt;/code&gt;&quot;) or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add emoji...&lt;/i&gt;]]&quot;
				)
			)
		)
	-- SECTION: Scaling Characteristics
	--[[ Some data will be collected first. ]]
		-- (Base) Behavior
	local features = {}
	local starts = {}
	local scalings = {}
	if (featureCount &gt; 1) then
		for i = 1, featureCount do
			table.insert(features, 
				{
					frame.args[&quot;Feature &quot;..i], 
					&quot; (&quot; .. p.var(frame.args[&quot;Var &quot;..i]) .. &quot;)&quot;
				})
			table.insert(starts, 
				{
					p.var( frame.args[&quot;Var0 &quot;..i] or (frame.args[&quot;Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;, 
					frame.args[&quot;Starting value &quot;..i]
				})
			table.insert(scalings,
				{
					p.var( &quot;Δ&quot;..(frame.args[&quot;Var &quot;..i]) ) .. &quot;: &quot;,
					frame.args[&quot;Scaling &quot;..i]
				})
		end
	else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1
		features[1] = {frame.args[&quot;Feature&quot;],&quot;&quot;}
		starts[1] = {&quot;&quot;,frame.args[&quot;Starting value&quot;]}
		scalings[1] = {&quot;&quot;,frame.args[&quot;Scaling&quot;]}
	end
		-- Super Behavior
	local s_features
	local s_starts
	local s_scalings
	-- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default.
	local has_s_features;
	local has_s_starts;
	local has_s_scalings;
	if (frame.args[&quot;SUPER Feature count&quot;]) then
		if s_featureCount &gt; 1 then 
			-- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data
			s_features = mw.clone(features)
			s_starts = mw.clone(starts)
			s_scalings = mw.clone(scalings)
			-- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors.
			if s_featureCount &gt; featureCount then for i = featureCount + 1, s_featureCount do 
					table.insert( s_features, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_starts, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_scalings, {&quot;&quot;,&quot;&quot;} )
				end
			elseif s_featureCount &lt; featureCount then for i = s_featureCount + 1, featureCount do 
					s_features[i] = nil
					s_starts[i] = nil
					s_scalings[i] = nil
				end
			end
			-- Parse super data, set the has_s_... flags if triggered
			for i = 1,s_featureCount do
				if frame.args[&quot;SUPER Feature &quot;..i] then 
					s_features[i][1] = frame.args[&quot;SUPER Feature &quot;..i] 
					if (not has_s_features) then has_s_features = true end
				end
				if frame.args[&quot;SUPER Starting value &quot;..i] then 
					s_starts[i][2] = frame.args[&quot;SUPER Starting value &quot;..i] 
					if (not has_s_starts) then has_s_starts = true end
				end
				if frame.args[&quot;SUPER Scaling &quot;..i] then 
					s_scalings[i][2] = frame.args[&quot;SUPER Scaling &quot;..i] 
					if (not has_s_scalings) then has_s_scalings = true end
				end
				if frame.args[&quot;SUPER Var &quot;..i] then
					s_features[i][2] = &quot; (&quot; .. p.var(frame.args[&quot;SUPER Var &quot;..i]) .. &quot;)&quot;
					s_starts[i][1] = p.var( frame.args[&quot;SUPER Var0 &quot;..i] or (frame.args[&quot;SUPER Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;
					s_scalings[i][1] = p.var( '&lt;span style=&quot;font-style:normal&quot;&gt;Δ&lt;/span&gt;'..(frame.args[&quot;SUPER Var &quot;..i]) ) .. &quot;: &quot;
					if (not has_s_features) then has_s_features = true end
					if (not has_s_starts) then has_s_starts = true end
					if (not has_s_scalings) then has_s_scalings = true end
				end 
				s_features[i] = table.concat(s_features[i])
				s_starts[i] = table.concat(s_starts[i])
				s_scalings[i] = table.concat(s_scalings[i])
			end
			s_features = table.concat(s_features,&quot;&lt;br&gt;&quot;)
			s_starts = table.concat(s_starts,&quot;&lt;br&gt;&quot;)
			s_scalings = table.concat(s_scalings,&quot;&lt;br&gt;&quot;)
		else -- # of super features = 1? just write directly to the variable
			if frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Feature&quot;] then 
				has_s_features = true
				s_features = frame.args[&quot;SUPER Feature&quot;]
			end
			if frame.args[&quot;SUPER Var0&quot;] or frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Starting value&quot;] then 
				has_s_starts = true
				s_starts = frame.args[&quot;SUPER Starting value&quot;]
			end
			if frame.args[&quot;SUPER Scaling&quot;] then 
				has_s_scalings = true
				s_scalings = frame.args[&quot;SUPER Scaling&quot;]
			end
		end
	end
		-- finalize base data as strings now that work is otherwise complete
	if (featureCount &gt; 1) then 
		for i = 1, featureCount do
			features[i] = table.concat(features[i])
			starts[i] = table.concat(starts[i])
			scalings[i] = table.concat(scalings[i])
		end 
		features = table.concat(features,&quot;&lt;br&gt;&quot;)
		starts = table.concat(starts,&quot;&lt;br&gt;&quot;)
		scalings = table.concat(scalings,&quot;&lt;br&gt;&quot;)
	else
		features = features[1][1]
		starts = starts[1][2]
		scalings = scalings[1][2]
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (frame.args[&quot;SUPER Feature count&quot;]), &quot;Base Scaling Characteristics&quot;, &quot;Scaling Characteristics&quot; ) )
		:node( (frame.args[&quot;SUPER Feature count&quot;]) and p.s_subheading( &quot;Super Scaling Characteristics&quot; ) )
		-- Features
		:node( p.row( (has_s_features), 
				(featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			features
		) )
		:node( (has_s_features) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;SUPER Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;SUPER Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			s_features
		) )
		-- Starting values
		:node( p.row( (has_s_starts), 
				(featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;Var0&quot;] or (frame.args[&quot;Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			starts
		) )
		:node( (has_s_starts) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;SUPER Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;SUPER Var0&quot;] or (frame.args[&quot;SUPER Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			s_starts
		) )
		-- Scaling
		:node( p.row( (has_s_scalings), &quot;Scaling&quot;, scalings) )
		:node( (has_s_scalings) and p.s_row(&quot;Scaling&quot;, s_scalings) )
	-- SECTION: Career 
	--[[ Some data will be collected first. ]]
		-- (Base) Games played
	local gamesPlayedText = {}
	if matchCount.total &gt; 0 then
		table.insert(gamesPlayedText, matchCount.total .. &quot; (total)&quot;)
		if matchCount.standard &gt; 0 then table.insert(gamesPlayedText, matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- (Base) Winrate
	local winrateText = {}
	if matchCount.total &gt; 0 then
		if matchCount.standard &gt; 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
		-- Super Games played
	local s_gamesPlayedText = {}
	if s_matchCount.total &gt; 0 then
		table.insert(s_gamesPlayedText, s_matchCount.total .. &quot; (total)&quot;)
		if s_matchCount.standard &gt; 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- Super Winrate
	local s_winrateText = {}
	if s_matchCount.total &gt; 0 then
		if s_matchCount.standard &gt; 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (isSuper), &quot;Base Career&quot;, &quot;Career &quot;) )
		:node( (isSuper) and p.s_subheading( &quot;Super Career&quot;) )
		-- Debuted
		:node( p.row( (isSuper), &quot;Debut&quot;, frame.args[&quot;Debut&quot;] ) )
		:node( (isSuper) and p.s_row(&quot;Debut&quot;, frame.args[&quot;SUPER Debut&quot;]) )
		-- Games played
		:node( p.row( 
			(isSuper), 
			&quot;Games played&quot;, 
				(#gamesPlayedText &gt; 0) and table.concat(gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot; 
		) )
		:node( (isSuper) and p.s_row(
			&quot;Games played&quot;, 
				(#s_gamesPlayedText &gt; 0) and table.concat(s_gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		-- Winrate
		:node( p.row( 
			(isSuper), 
			&quot;Winrate&quot;, 
				(#winrateText &gt; 0) and table.concat(winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		:node( (isSuper) and p.s_row(
			&quot;Winrate&quot;, 
				(#s_winrateText &gt; 0) and table.concat(s_winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
	return tostring(infobox)
end

function p.referenceTest()
	local element = mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible&quot; )
		:css( &quot;text-align&quot;,&quot;left&quot; )

	local bigElement = mw.html.create( &quot;div&quot; )
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface&quot;)
		)
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface2&quot;)
		)

	return tostring(element) .. &quot;\n&quot; .. tostring(bigElement) .. &quot;\n&quot;
	
end

function p.roundedPct(num, den)
	return (num/den % 0.01 &lt; 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100)
end

function p.title()
	--[[local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;
		}
	}
	local cases2 = &quot;ba&quot;
	local cases3 = &quot;ba&quot;
	-- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = &quot;true&quot; else cases2 = &quot;false&quot; end
	-- if &quot;ANCOVA&quot; == frame:getParent():getTitle() then cases3 = &quot;true&quot; else cases3 = &quot;false&quot; end
	local detected = false
	detected = detected or true]]
	return mw.title.getCurrentTitle() 
end

function p.ifexist()
	return tonumber(frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot;, args = {frame.args[1]} })
end

function p.performance()
	for i = 1,5000000 do p.tableMake() end
	return
end

function p.performance1()
	local var = &quot;&quot;
	for i = 1,100000000 do if var[1] then end end
	return
end

function p.performance2()
	local var = &quot;&quot;
	for i = 1,100000000 do if type(var) == &quot;table&quot; then end end
	return
end

p.counter = 0
function p.tableBake()
	p.counter = p.counter + 1
	return p.counter
end


function p.tableMake()
	p.counter = 0
	local var = {
		keyword = {p.tableBake()},
		kljuczslovo = {p.tableBake()},
		yaoshiwenzi = {p.tableBake()},
		llavepalabra = {p.tableBake()},
	}
	return var
end

function p.expand()
	return frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot; }
end

function p.title3()
	return mw.title.getCurrentTitle().fullText
end

function p.getBallAttribute()
	local _, ballData = pcall( p.getBallAttributeInternal, frame.args[1], frame.args[2], frame.args[3] )
	if _ then return ballData
	else return &quot;&quot; end
end

function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att]
	if ballData[1] then
		return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)]
	end
	return ballData
end

function p.emojiCodepointInternal( emoji )
	emoji = mw.text.trim( emoji )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.emojiCodepoint()
	local emoji = mw.text.trim( mw.getCurrentFrame().args[1] )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.title2()
	local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;,
			[&quot;Module:Sandbox/22&quot;] = &quot;Module:Sandbox/22&quot;
		}
	}
	
	local concepts = {
		grammatical_cases,
	}
	local PAGENAME = tostring(mw.title.getCurrentTitle())
	-- local detected = false
	local out = {}
	local debugger = {}
	for i, group in ipairs(concepts) do 
		if group.inessive[PAGENAME] then 
			table.insert(debugger,&quot;Okay, managed TRUE on &quot;..i..&quot;:&quot;..group.ablative)
			-- detected = detected or true
			local formattedGroup = {}
			for memberKey, memberRendervalue in pairs(group.inessive) do
				table.insert(formattedGroup, &quot;&lt;li&gt;&quot; .. memberRendervalue .. &quot;&lt;/li&gt;&quot;)
			end 
			table.insert(out, &quot;&lt;h3&gt;&quot; .. group.ablative .. &quot;&lt;/h3&gt;&lt;ul&gt;&quot; .. table.concat( formattedGroup ) .. &quot;&lt;/ul&gt;&quot; )
			table.insert(out, &quot;this sucks ass&quot;)
			return &quot;this sucks ass&quot;
		else
			table.insert(debugger,&quot;Managed FALSE on &quot;..i..&quot;:&quot;..group.ablative..&quot;. Comparator: &quot;..PAGENAME..&quot;, &quot;)
		end
	end
	return debugger
	--[[
	if detected then return tostring(detected) .. table.concat(out) end
	return
	--]]
end

function p.sanitizeChar(capture)
	-- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited
	if capture == &quot;?&quot; then
		return &quot;&quot;
	elseif capture == &quot;#&quot; then
		return &quot;No. &quot;
	end
end

function p.sanitizeTitle()
	local title = frame.args[1]
	return mw.ustring.gsub( title, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar) 
end

function p.sanitizeTitletest2()
	return mw.ustring.gsub( &quot;Ki vagy?  (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)&quot;, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar()) 
end

function p.safetyTest()
	if string.len(frame.args[1]) &lt; 500 then 
		return frame.args[1] 
	else 
		return &quot;&quot;
	end
end

--[[ 
function array_iter(t)
 	local i = 0
	return function ()
		i = i + 1
		return t[i] 
	end 
end
--]]

function p.listInline()
	--[[ This function is a godsend! 
	https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]]
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return mw.text.listToText( spritedTable , &quot;, &quot;, &quot;, and &quot; )
end

function p.listNavbox()
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return table.concat( spritedTable , &quot; • &quot;)
end

function p.listCountInternal(table) 
	local count = 0
	for index,value in ipairs( table ) do count = count + 1 end
	return count
end

function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn

function p.tableTest()
	return balls[1][1]
end

function p.todaysBall()
	--[[ https://www.lua.org/pil/3.6.html
		 The use of explicit indexing here is not strictly semantic;
		 it's just to emphasize the rotation order.
	 ]]
	local unixDay = math.floor(os.time()/86400)
	local ballTotalCount = 0
	local ballCounts = {}
    for whichTable,subtable in ipairs(balls) do 
		ballCounts[whichTable] = #subtable
		ballTotalCount = ballTotalCount + ballCounts[whichTable]
	end
	local cycleDayIndex,  cycleStartingBall = unixDay % ballTotalCount,  math.floor(unixDay/ballTotalCount) % ballTotalCount
	local cycleTodaysBall
	if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount
	elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount
	elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount
	else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end
	--[[ debug
	return cycleTodaysBall .. &quot; &quot; .. ballTotalCount .. &quot; &quot; .. os.time() .. &quot; &quot; .. math.floor(os.time()/86400) .. &quot; &quot; .. cycleDayIndex .. &quot; &quot; .. cycleStartingBall
	]]
	for whichTable = 1, #balls do 
		if ballCounts[whichTable] &gt; cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] 
		else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end
	end 
	--]]
end

return p;</text>
      <sha1>sla1koklbc9jehl6uk456n7cxmyesnq</sha1>
    </revision>
    <revision>
      <id>2834742</id>
      <parentid>2834741</parentid>
      <timestamp>2026-09-28T00:16:11Z</timestamp>
      <contributor>
        <username>Helpme2222</username>
        <id>3106525</id>
      </contributor>
      <origin>2834742</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="20868" sha1="ql494wda2uan5884rdx7iyk5t30hduh" xml:space="preserve">-- My function for teaching grammar

local p = {};
local frame = mw.getCurrentFrame()
local grammatical_moods = {
	&quot;Jelen ido&quot;,
	&quot;Mult ido&quot;,
	&quot;Felszolito mod&quot;,
	&quot;Felteteles mod&quot;
}
local balls = {
	grammatical_moods
}

local toggleButton = mw.html.create( 'span' )
	:css({
		float = &quot;right&quot;,
		[&quot;margin-right&quot;] = &quot;0&quot;,
		[&quot;font-weight&quot;] = &quot;normal&quot;
	})
	:addClass(&quot;mw-customtoggle-Infobox&quot;)
	:node(&quot;[Click to toggle]&quot;)

function p.subheading( discrim, b_content, content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node((discrim) and b_content or content)
			:node((discrim) and toggleButton)
		)
end
	
function p.s_subheading( s_content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible mw-collapsed&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node(s_content)
			:node(toggleButton)
		)
end

function p.row( discrim, name, data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; ) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(name)
		)
		:node( mw.html.create( 'td' )
			:node(data)
		)
end
	
function p.s_row( s_name, s_data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass(&quot;mw-collapsible mw-collapsed&quot;) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(s_name)
		)
		:node( mw.html.create( 'td' )
			:node(s_data)
		)
end

function p.var(symbol)
	return frame:expandTemplate{ title = &quot;Template:Var&quot;, args = {symbol} }
end
	

function p.infoboxWBB()
	-- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html
	--[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node 
			Thankfully, these operations interpret passed nils as no-ops 
			(I'm technically using false though. Somehow that works anyways?)]]
	
	local frame = mw.getCurrentFrame()
	local ballname = frame.args[&quot;Title&quot;] or tostring(mw.title.getCurrentTitle())
	local Entities = p -- require(&quot;Module:Entities&quot;)
	
	local featureCount = tonumber(frame.args[&quot;Feature count&quot;]) or 1
	local s_featureCount = tonumber(frame.args[&quot;SUPER Feature count&quot;]) or featureCount
	local matchCount = {
		total = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local matchWinCount = {
		standard = mw.site.stats.pagesInCategory( ballname..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local s_matchCount = {
		total = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local s_matchWinCount = {
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local isSuper = 
		frame.args[&quot;SUPER Feature count&quot;] or
		frame.args[&quot;SUPER Type&quot;] or 
		frame.args[&quot;SUPER Debuted&quot;] or
		(s_matchCount.total &gt; 0) or 
		nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra &quot;or nil&quot; had to be added since this is used in :addClass calls.
	-- TITLE
	local infobox = mw.html.create( 'table' )
		:addClass(&quot;wikitable&quot;)
		:css({
			float = &quot;right&quot;; 
			[&quot;margin-left&quot;] = &quot;10px&quot;
		})
		:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass((isSuper) and &quot;mw-collapsible&quot; or nil)
				:node(&quot;&lt;big&gt;&quot; .. ballname .. &quot;&lt;/big&gt;&quot;)
			)
		)
		:node((isSuper) and mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass(&quot;mw-collapsible mw-collapsed&quot;)
				:node(&quot;&lt;big&gt;&quot; .. (frame.args[&quot;SUPER Title&quot;] or (&quot;Super &quot; .. ballname)) .. &quot;&lt;/big&gt;&quot;)
			)
		)
	-- SECTION: Identification
		-- Section Heading
		:node( p.subheading( (frame.args[&quot;SUPER Type&quot;]), &quot;Base Identification&quot;, &quot;Basic Info&quot; ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_subheading( &quot;Super Identification&quot; ) )
		-- Type
		:node( p.row( (frame.args[&quot;SUPER Type&quot;]), &quot;Type&quot;, frame.args[&quot;Type&quot;] ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_row( &quot;Type&quot;, frame.args[&quot;SUPER Type&quot;] ) )
		-- Color
	local color = frame.args[&quot;Color&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;color&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Color&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(color ~= &quot;&quot;) and frame:expandTemplate{ title = &quot;Color&quot;, args = {color} } or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add color...&lt;/i&gt;]]&quot;
				)
			)
		)
		-- Emoji
	local emoji = frame.args[&quot;Emoji&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;emoji&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Emoji&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(emoji ~= &quot;&quot;) and ( emoji .. &quot; &lt;code&gt;&quot; .. p.emojiCodepointInternal(emoji) .. &quot;&lt;/code&gt;&quot;) or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add emoji...&lt;/i&gt;]]&quot;
				)
			)
		)
	-- SECTION: Scaling Characteristics
	--[[ Some data will be collected first. ]]
		-- (Base) Behavior
	local features = {}
	local starts = {}
	local scalings = {}
	if (featureCount &gt; 1) then
		for i = 1, featureCount do
			table.insert(features, 
				{
					frame.args[&quot;Feature &quot;..i], 
					&quot; (&quot; .. p.var(frame.args[&quot;Var &quot;..i]) .. &quot;)&quot;
				})
			table.insert(starts, 
				{
					p.var( frame.args[&quot;Var0 &quot;..i] or (frame.args[&quot;Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;, 
					frame.args[&quot;Starting value &quot;..i]
				})
			table.insert(scalings,
				{
					p.var( &quot;Δ&quot;..(frame.args[&quot;Var &quot;..i]) ) .. &quot;: &quot;,
					frame.args[&quot;Scaling &quot;..i]
				})
		end
	else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1
		features[1] = {frame.args[&quot;Feature&quot;],&quot;&quot;}
		starts[1] = {&quot;&quot;,frame.args[&quot;Starting value&quot;]}
		scalings[1] = {&quot;&quot;,frame.args[&quot;Scaling&quot;]}
	end
		-- Super Behavior
	local s_features
	local s_starts
	local s_scalings
	-- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default.
	local has_s_features;
	local has_s_starts;
	local has_s_scalings;
	if (frame.args[&quot;SUPER Feature count&quot;]) then
		if s_featureCount &gt; 1 then 
			-- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data
			s_features = mw.clone(features)
			s_starts = mw.clone(starts)
			s_scalings = mw.clone(scalings)
			-- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors.
			if s_featureCount &gt; featureCount then for i = featureCount + 1, s_featureCount do 
					table.insert( s_features, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_starts, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_scalings, {&quot;&quot;,&quot;&quot;} )
				end
			elseif s_featureCount &lt; featureCount then for i = s_featureCount + 1, featureCount do 
					s_features[i] = nil
					s_starts[i] = nil
					s_scalings[i] = nil
				end
			end
			-- Parse super data, set the has_s_... flags if triggered
			for i = 1,s_featureCount do
				if frame.args[&quot;SUPER Feature &quot;..i] then 
					s_features[i][1] = frame.args[&quot;SUPER Feature &quot;..i] 
					if (not has_s_features) then has_s_features = true end
				end
				if frame.args[&quot;SUPER Starting value &quot;..i] then 
					s_starts[i][2] = frame.args[&quot;SUPER Starting value &quot;..i] 
					if (not has_s_starts) then has_s_starts = true end
				end
				if frame.args[&quot;SUPER Scaling &quot;..i] then 
					s_scalings[i][2] = frame.args[&quot;SUPER Scaling &quot;..i] 
					if (not has_s_scalings) then has_s_scalings = true end
				end
				if frame.args[&quot;SUPER Var &quot;..i] then
					s_features[i][2] = &quot; (&quot; .. p.var(frame.args[&quot;SUPER Var &quot;..i]) .. &quot;)&quot;
					s_starts[i][1] = p.var( frame.args[&quot;SUPER Var0 &quot;..i] or (frame.args[&quot;SUPER Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;
					s_scalings[i][1] = p.var( '&lt;span style=&quot;font-style:normal&quot;&gt;Δ&lt;/span&gt;'..(frame.args[&quot;SUPER Var &quot;..i]) ) .. &quot;: &quot;
					if (not has_s_features) then has_s_features = true end
					if (not has_s_starts) then has_s_starts = true end
					if (not has_s_scalings) then has_s_scalings = true end
				end 
				s_features[i] = table.concat(s_features[i])
				s_starts[i] = table.concat(s_starts[i])
				s_scalings[i] = table.concat(s_scalings[i])
			end
			s_features = table.concat(s_features,&quot;&lt;br&gt;&quot;)
			s_starts = table.concat(s_starts,&quot;&lt;br&gt;&quot;)
			s_scalings = table.concat(s_scalings,&quot;&lt;br&gt;&quot;)
		else -- # of super features = 1? just write directly to the variable
			if frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Feature&quot;] then 
				has_s_features = true
				s_features = frame.args[&quot;SUPER Feature&quot;]
			end
			if frame.args[&quot;SUPER Var0&quot;] or frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Starting value&quot;] then 
				has_s_starts = true
				s_starts = frame.args[&quot;SUPER Starting value&quot;]
			end
			if frame.args[&quot;SUPER Scaling&quot;] then 
				has_s_scalings = true
				s_scalings = frame.args[&quot;SUPER Scaling&quot;]
			end
		end
	end
		-- finalize base data as strings now that work is otherwise complete
	if (featureCount &gt; 1) then 
		for i = 1, featureCount do
			features[i] = table.concat(features[i])
			starts[i] = table.concat(starts[i])
			scalings[i] = table.concat(scalings[i])
		end 
		features = table.concat(features,&quot;&lt;br&gt;&quot;)
		starts = table.concat(starts,&quot;&lt;br&gt;&quot;)
		scalings = table.concat(scalings,&quot;&lt;br&gt;&quot;)
	else
		features = features[1][1]
		starts = starts[1][2]
		scalings = scalings[1][2]
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (frame.args[&quot;SUPER Feature count&quot;]), &quot;Base Scaling Characteristics&quot;, &quot;Scaling Characteristics&quot; ) )
		:node( (frame.args[&quot;SUPER Feature count&quot;]) and p.s_subheading( &quot;Super Scaling Characteristics&quot; ) )
		-- Features
		:node( p.row( (has_s_features), 
				(featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			features
		) )
		:node( (has_s_features) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;SUPER Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;SUPER Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			s_features
		) )
		-- Starting values
		:node( p.row( (has_s_starts), 
				(featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;Var0&quot;] or (frame.args[&quot;Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			starts
		) )
		:node( (has_s_starts) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;SUPER Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;SUPER Var0&quot;] or (frame.args[&quot;SUPER Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			s_starts
		) )
		-- Scaling
		:node( p.row( (has_s_scalings), &quot;Scaling&quot;, scalings) )
		:node( (has_s_scalings) and p.s_row(&quot;Scaling&quot;, s_scalings) )
	-- SECTION: Career 
	--[[ Some data will be collected first. ]]
		-- (Base) Games played
	local gamesPlayedText = {}
	if matchCount.total &gt; 0 then
		table.insert(gamesPlayedText, matchCount.total .. &quot; (total)&quot;)
		if matchCount.standard &gt; 0 then table.insert(gamesPlayedText, matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- (Base) Winrate
	local winrateText = {}
	if matchCount.total &gt; 0 then
		if matchCount.standard &gt; 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
		-- Super Games played
	local s_gamesPlayedText = {}
	if s_matchCount.total &gt; 0 then
		table.insert(s_gamesPlayedText, s_matchCount.total .. &quot; (total)&quot;)
		if s_matchCount.standard &gt; 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- Super Winrate
	local s_winrateText = {}
	if s_matchCount.total &gt; 0 then
		if s_matchCount.standard &gt; 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (isSuper), &quot;Base Career&quot;, &quot;Career &quot;) )
		:node( (isSuper) and p.s_subheading( &quot;Super Career&quot;) )
		-- Debuted
		:node( p.row( (isSuper), &quot;Debut&quot;, frame.args[&quot;Debut&quot;] ) )
		:node( (isSuper) and p.s_row(&quot;Debut&quot;, frame.args[&quot;SUPER Debut&quot;]) )
		-- Games played
		:node( p.row( 
			(isSuper), 
			&quot;Games played&quot;, 
				(#gamesPlayedText &gt; 0) and table.concat(gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot; 
		) )
		:node( (isSuper) and p.s_row(
			&quot;Games played&quot;, 
				(#s_gamesPlayedText &gt; 0) and table.concat(s_gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		-- Winrate
		:node( p.row( 
			(isSuper), 
			&quot;Winrate&quot;, 
				(#winrateText &gt; 0) and table.concat(winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		:node( (isSuper) and p.s_row(
			&quot;Winrate&quot;, 
				(#s_winrateText &gt; 0) and table.concat(s_winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
	return tostring(infobox)
end

function p.referenceTest()
	local element = mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible&quot; )
		:css( &quot;text-align&quot;,&quot;left&quot; )

	local bigElement = mw.html.create( &quot;div&quot; )
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface&quot;)
		)
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface2&quot;)
		)

	return tostring(element) .. &quot;\n&quot; .. tostring(bigElement) .. &quot;\n&quot;
	
end

function p.roundedPct(num, den)
	return (num/den % 0.01 &lt; 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100)
end

function p.title()
	--[[local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;
		}
	}
	local cases2 = &quot;ba&quot;
	local cases3 = &quot;ba&quot;
	-- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = &quot;true&quot; else cases2 = &quot;false&quot; end
	-- if &quot;ANCOVA&quot; == frame:getParent():getTitle() then cases3 = &quot;true&quot; else cases3 = &quot;false&quot; end
	local detected = false
	detected = detected or true]]
	return mw.title.getCurrentTitle() 
end

function p.ifexist()
	return tonumber(frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot;, args = {frame.args[1]} })
end

function p.performance()
	for i = 1,5000000 do p.tableMake() end
	return
end

function p.performance1()
	local var = &quot;&quot;
	for i = 1,100000000 do if var[1] then end end
	return
end

function p.performance2()
	local var = &quot;&quot;
	for i = 1,100000000 do if type(var) == &quot;table&quot; then end end
	return
end

p.counter = 0
function p.tableBake()
	p.counter = p.counter + 1
	return p.counter
end


function p.tableMake()
	p.counter = 0
	local var = {
		keyword = {p.tableBake()},
		kljuczslovo = {p.tableBake()},
		yaoshiwenzi = {p.tableBake()},
		llavepalabra = {p.tableBake()},
	}
	return var
end

function p.expand()
	return frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot; }
end

function p.title3()
	return mw.title.getCurrentTitle().fullText
end

function p.getBallAttribute()
	local _, ballData = pcall( p.getBallAttributeInternal, frame.args[1], frame.args[2], frame.args[3], frame.args[4] )
	if _ then return ballData 
	else return &quot;&quot; end
end

function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att]
	if ballData[1] then
		return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)]
	end
	return ballData
end

function p.emojiCodepointInternal( emoji )
	emoji = mw.text.trim( emoji )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.emojiCodepoint()
	local emoji = mw.text.trim( mw.getCurrentFrame().args[1] )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.title2()
	local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;,
			[&quot;Module:Sandbox/22&quot;] = &quot;Module:Sandbox/22&quot;
		}
	}
	
	local concepts = {
		grammatical_cases,
	}
	local PAGENAME = tostring(mw.title.getCurrentTitle())
	-- local detected = false
	local out = {}
	local debugger = {}
	for i, group in ipairs(concepts) do 
		if group.inessive[PAGENAME] then 
			table.insert(debugger,&quot;Okay, managed TRUE on &quot;..i..&quot;:&quot;..group.ablative)
			-- detected = detected or true
			local formattedGroup = {}
			for memberKey, memberRendervalue in pairs(group.inessive) do
				table.insert(formattedGroup, &quot;&lt;li&gt;&quot; .. memberRendervalue .. &quot;&lt;/li&gt;&quot;)
			end 
			table.insert(out, &quot;&lt;h3&gt;&quot; .. group.ablative .. &quot;&lt;/h3&gt;&lt;ul&gt;&quot; .. table.concat( formattedGroup ) .. &quot;&lt;/ul&gt;&quot; )
			table.insert(out, &quot;this sucks ass&quot;)
			return &quot;this sucks ass&quot;
		else
			table.insert(debugger,&quot;Managed FALSE on &quot;..i..&quot;:&quot;..group.ablative..&quot;. Comparator: &quot;..PAGENAME..&quot;, &quot;)
		end
	end
	return debugger
	--[[
	if detected then return tostring(detected) .. table.concat(out) end
	return
	--]]
end

function p.sanitizeChar(capture)
	-- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited
	if capture == &quot;?&quot; then
		return &quot;&quot;
	elseif capture == &quot;#&quot; then
		return &quot;No. &quot;
	end
end

function p.sanitizeTitle()
	local title = frame.args[1]
	return mw.ustring.gsub( title, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar) 
end

function p.sanitizeTitletest2()
	return mw.ustring.gsub( &quot;Ki vagy?  (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)&quot;, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar()) 
end

function p.safetyTest()
	if string.len(frame.args[1]) &lt; 500 then 
		return frame.args[1] 
	else 
		return &quot;&quot;
	end
end

--[[ 
function array_iter(t)
 	local i = 0
	return function ()
		i = i + 1
		return t[i] 
	end 
end
--]]

function p.listInline()
	--[[ This function is a godsend! 
	https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]]
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return mw.text.listToText( spritedTable , &quot;, &quot;, &quot;, and &quot; )
end

function p.listNavbox()
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return table.concat( spritedTable , &quot; • &quot;)
end

function p.listCountInternal(table) 
	local count = 0
	for index,value in ipairs( table ) do count = count + 1 end
	return count
end

function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn

function p.tableTest()
	return balls[1][1]
end

function p.todaysBall()
	--[[ https://www.lua.org/pil/3.6.html
		 The use of explicit indexing here is not strictly semantic;
		 it's just to emphasize the rotation order.
	 ]]
	local unixDay = math.floor(os.time()/86400)
	local ballTotalCount = 0
	local ballCounts = {}
    for whichTable,subtable in ipairs(balls) do 
		ballCounts[whichTable] = #subtable
		ballTotalCount = ballTotalCount + ballCounts[whichTable]
	end
	local cycleDayIndex,  cycleStartingBall = unixDay % ballTotalCount,  math.floor(unixDay/ballTotalCount) % ballTotalCount
	local cycleTodaysBall
	if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount
	elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount
	elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount
	else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end
	--[[ debug
	return cycleTodaysBall .. &quot; &quot; .. ballTotalCount .. &quot; &quot; .. os.time() .. &quot; &quot; .. math.floor(os.time()/86400) .. &quot; &quot; .. cycleDayIndex .. &quot; &quot; .. cycleStartingBall
	]]
	for whichTable = 1, #balls do 
		if ballCounts[whichTable] &gt; cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] 
		else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end
	end 
	--]]
end

return p;</text>
      <sha1>ql494wda2uan5884rdx7iyk5t30hduh</sha1>
    </revision>
    <revision>
      <id>2834743</id>
      <parentid>2834742</parentid>
      <timestamp>2026-09-28T00:27:03Z</timestamp>
      <contributor>
        <username>Helpme2222</username>
        <id>3106525</id>
      </contributor>
      <origin>2834743</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="21067" sha1="gxwxollrdv4irgbqm2d8pg9f0oq5bqx" xml:space="preserve">-- My function for teaching grammar

local p = {};
local frame = mw.getCurrentFrame()
local grammatical_moods = {
	&quot;Jelen ido&quot;,
	&quot;Mult ido&quot;,
	&quot;Felszolito mod&quot;,
	&quot;Felteteles mod&quot;
}
local balls = {
	grammatical_moods
}

local toggleButton = mw.html.create( 'span' )
	:css({
		float = &quot;right&quot;,
		[&quot;margin-right&quot;] = &quot;0&quot;,
		[&quot;font-weight&quot;] = &quot;normal&quot;
	})
	:addClass(&quot;mw-customtoggle-Infobox&quot;)
	:node(&quot;[Click to toggle]&quot;)

function p.subheading( discrim, b_content, content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node((discrim) and b_content or content)
			:node((discrim) and toggleButton)
		)
end
	
function p.s_subheading( s_content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible mw-collapsed&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node(s_content)
			:node(toggleButton)
		)
end

function p.row( discrim, name, data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; ) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(name)
		)
		:node( mw.html.create( 'td' )
			:node(data)
		)
end
	
function p.s_row( s_name, s_data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass(&quot;mw-collapsible mw-collapsed&quot;) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(s_name)
		)
		:node( mw.html.create( 'td' )
			:node(s_data)
		)
end

function p.var(symbol)
	return frame:expandTemplate{ title = &quot;Template:Var&quot;, args = {symbol} }
end
	

function p.infoboxWBB()
	-- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html
	--[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node 
			Thankfully, these operations interpret passed nils as no-ops 
			(I'm technically using false though. Somehow that works anyways?)]]
	
	local frame = mw.getCurrentFrame()
	local ballname = frame.args[&quot;Title&quot;] or tostring(mw.title.getCurrentTitle())
	local Entities = p -- require(&quot;Module:Entities&quot;)
	
	local featureCount = tonumber(frame.args[&quot;Feature count&quot;]) or 1
	local s_featureCount = tonumber(frame.args[&quot;SUPER Feature count&quot;]) or featureCount
	local matchCount = {
		total = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local matchWinCount = {
		standard = mw.site.stats.pagesInCategory( ballname..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local s_matchCount = {
		total = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local s_matchWinCount = {
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local isSuper = 
		frame.args[&quot;SUPER Feature count&quot;] or
		frame.args[&quot;SUPER Type&quot;] or 
		frame.args[&quot;SUPER Debuted&quot;] or
		(s_matchCount.total &gt; 0) or 
		nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra &quot;or nil&quot; had to be added since this is used in :addClass calls.
	-- TITLE
	local infobox = mw.html.create( 'table' )
		:addClass(&quot;wikitable&quot;)
		:css({
			float = &quot;right&quot;; 
			[&quot;margin-left&quot;] = &quot;10px&quot;
		})
		:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass((isSuper) and &quot;mw-collapsible&quot; or nil)
				:node(&quot;&lt;big&gt;&quot; .. ballname .. &quot;&lt;/big&gt;&quot;)
			)
		)
		:node((isSuper) and mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass(&quot;mw-collapsible mw-collapsed&quot;)
				:node(&quot;&lt;big&gt;&quot; .. (frame.args[&quot;SUPER Title&quot;] or (&quot;Super &quot; .. ballname)) .. &quot;&lt;/big&gt;&quot;)
			)
		)
	-- SECTION: Identification
		-- Section Heading
		:node( p.subheading( (frame.args[&quot;SUPER Type&quot;]), &quot;Base Identification&quot;, &quot;Basic Info&quot; ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_subheading( &quot;Super Identification&quot; ) )
		-- Type
		:node( p.row( (frame.args[&quot;SUPER Type&quot;]), &quot;Type&quot;, frame.args[&quot;Type&quot;] ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_row( &quot;Type&quot;, frame.args[&quot;SUPER Type&quot;] ) )
		-- Color
	local color = frame.args[&quot;Color&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;color&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Color&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(color ~= &quot;&quot;) and frame:expandTemplate{ title = &quot;Color&quot;, args = {color} } or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add color...&lt;/i&gt;]]&quot;
				)
			)
		)
		-- Emoji
	local emoji = frame.args[&quot;Emoji&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;emoji&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Emoji&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(emoji ~= &quot;&quot;) and ( emoji .. &quot; &lt;code&gt;&quot; .. p.emojiCodepointInternal(emoji) .. &quot;&lt;/code&gt;&quot;) or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add emoji...&lt;/i&gt;]]&quot;
				)
			)
		)
	-- SECTION: Scaling Characteristics
	--[[ Some data will be collected first. ]]
		-- (Base) Behavior
	local features = {}
	local starts = {}
	local scalings = {}
	if (featureCount &gt; 1) then
		for i = 1, featureCount do
			table.insert(features, 
				{
					frame.args[&quot;Feature &quot;..i], 
					&quot; (&quot; .. p.var(frame.args[&quot;Var &quot;..i]) .. &quot;)&quot;
				})
			table.insert(starts, 
				{
					p.var( frame.args[&quot;Var0 &quot;..i] or (frame.args[&quot;Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;, 
					frame.args[&quot;Starting value &quot;..i]
				})
			table.insert(scalings,
				{
					p.var( &quot;Δ&quot;..(frame.args[&quot;Var &quot;..i]) ) .. &quot;: &quot;,
					frame.args[&quot;Scaling &quot;..i]
				})
		end
	else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1
		features[1] = {frame.args[&quot;Feature&quot;],&quot;&quot;}
		starts[1] = {&quot;&quot;,frame.args[&quot;Starting value&quot;]}
		scalings[1] = {&quot;&quot;,frame.args[&quot;Scaling&quot;]}
	end
		-- Super Behavior
	local s_features
	local s_starts
	local s_scalings
	-- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default.
	local has_s_features;
	local has_s_starts;
	local has_s_scalings;
	if (frame.args[&quot;SUPER Feature count&quot;]) then
		if s_featureCount &gt; 1 then 
			-- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data
			s_features = mw.clone(features)
			s_starts = mw.clone(starts)
			s_scalings = mw.clone(scalings)
			-- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors.
			if s_featureCount &gt; featureCount then for i = featureCount + 1, s_featureCount do 
					table.insert( s_features, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_starts, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_scalings, {&quot;&quot;,&quot;&quot;} )
				end
			elseif s_featureCount &lt; featureCount then for i = s_featureCount + 1, featureCount do 
					s_features[i] = nil
					s_starts[i] = nil
					s_scalings[i] = nil
				end
			end
			-- Parse super data, set the has_s_... flags if triggered
			for i = 1,s_featureCount do
				if frame.args[&quot;SUPER Feature &quot;..i] then 
					s_features[i][1] = frame.args[&quot;SUPER Feature &quot;..i] 
					if (not has_s_features) then has_s_features = true end
				end
				if frame.args[&quot;SUPER Starting value &quot;..i] then 
					s_starts[i][2] = frame.args[&quot;SUPER Starting value &quot;..i] 
					if (not has_s_starts) then has_s_starts = true end
				end
				if frame.args[&quot;SUPER Scaling &quot;..i] then 
					s_scalings[i][2] = frame.args[&quot;SUPER Scaling &quot;..i] 
					if (not has_s_scalings) then has_s_scalings = true end
				end
				if frame.args[&quot;SUPER Var &quot;..i] then
					s_features[i][2] = &quot; (&quot; .. p.var(frame.args[&quot;SUPER Var &quot;..i]) .. &quot;)&quot;
					s_starts[i][1] = p.var( frame.args[&quot;SUPER Var0 &quot;..i] or (frame.args[&quot;SUPER Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;
					s_scalings[i][1] = p.var( '&lt;span style=&quot;font-style:normal&quot;&gt;Δ&lt;/span&gt;'..(frame.args[&quot;SUPER Var &quot;..i]) ) .. &quot;: &quot;
					if (not has_s_features) then has_s_features = true end
					if (not has_s_starts) then has_s_starts = true end
					if (not has_s_scalings) then has_s_scalings = true end
				end 
				s_features[i] = table.concat(s_features[i])
				s_starts[i] = table.concat(s_starts[i])
				s_scalings[i] = table.concat(s_scalings[i])
			end
			s_features = table.concat(s_features,&quot;&lt;br&gt;&quot;)
			s_starts = table.concat(s_starts,&quot;&lt;br&gt;&quot;)
			s_scalings = table.concat(s_scalings,&quot;&lt;br&gt;&quot;)
		else -- # of super features = 1? just write directly to the variable
			if frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Feature&quot;] then 
				has_s_features = true
				s_features = frame.args[&quot;SUPER Feature&quot;]
			end
			if frame.args[&quot;SUPER Var0&quot;] or frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Starting value&quot;] then 
				has_s_starts = true
				s_starts = frame.args[&quot;SUPER Starting value&quot;]
			end
			if frame.args[&quot;SUPER Scaling&quot;] then 
				has_s_scalings = true
				s_scalings = frame.args[&quot;SUPER Scaling&quot;]
			end
		end
	end
		-- finalize base data as strings now that work is otherwise complete
	if (featureCount &gt; 1) then 
		for i = 1, featureCount do
			features[i] = table.concat(features[i])
			starts[i] = table.concat(starts[i])
			scalings[i] = table.concat(scalings[i])
		end 
		features = table.concat(features,&quot;&lt;br&gt;&quot;)
		starts = table.concat(starts,&quot;&lt;br&gt;&quot;)
		scalings = table.concat(scalings,&quot;&lt;br&gt;&quot;)
	else
		features = features[1][1]
		starts = starts[1][2]
		scalings = scalings[1][2]
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (frame.args[&quot;SUPER Feature count&quot;]), &quot;Base Scaling Characteristics&quot;, &quot;Scaling Characteristics&quot; ) )
		:node( (frame.args[&quot;SUPER Feature count&quot;]) and p.s_subheading( &quot;Super Scaling Characteristics&quot; ) )
		-- Features
		:node( p.row( (has_s_features), 
				(featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			features
		) )
		:node( (has_s_features) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;SUPER Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;SUPER Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			s_features
		) )
		-- Starting values
		:node( p.row( (has_s_starts), 
				(featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;Var0&quot;] or (frame.args[&quot;Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			starts
		) )
		:node( (has_s_starts) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;SUPER Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;SUPER Var0&quot;] or (frame.args[&quot;SUPER Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			s_starts
		) )
		-- Scaling
		:node( p.row( (has_s_scalings), &quot;Scaling&quot;, scalings) )
		:node( (has_s_scalings) and p.s_row(&quot;Scaling&quot;, s_scalings) )
	-- SECTION: Career 
	--[[ Some data will be collected first. ]]
		-- (Base) Games played
	local gamesPlayedText = {}
	if matchCount.total &gt; 0 then
		table.insert(gamesPlayedText, matchCount.total .. &quot; (total)&quot;)
		if matchCount.standard &gt; 0 then table.insert(gamesPlayedText, matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- (Base) Winrate
	local winrateText = {}
	if matchCount.total &gt; 0 then
		if matchCount.standard &gt; 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
		-- Super Games played
	local s_gamesPlayedText = {}
	if s_matchCount.total &gt; 0 then
		table.insert(s_gamesPlayedText, s_matchCount.total .. &quot; (total)&quot;)
		if s_matchCount.standard &gt; 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- Super Winrate
	local s_winrateText = {}
	if s_matchCount.total &gt; 0 then
		if s_matchCount.standard &gt; 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (isSuper), &quot;Base Career&quot;, &quot;Career &quot;) )
		:node( (isSuper) and p.s_subheading( &quot;Super Career&quot;) )
		-- Debuted
		:node( p.row( (isSuper), &quot;Debut&quot;, frame.args[&quot;Debut&quot;] ) )
		:node( (isSuper) and p.s_row(&quot;Debut&quot;, frame.args[&quot;SUPER Debut&quot;]) )
		-- Games played
		:node( p.row( 
			(isSuper), 
			&quot;Games played&quot;, 
				(#gamesPlayedText &gt; 0) and table.concat(gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot; 
		) )
		:node( (isSuper) and p.s_row(
			&quot;Games played&quot;, 
				(#s_gamesPlayedText &gt; 0) and table.concat(s_gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		-- Winrate
		:node( p.row( 
			(isSuper), 
			&quot;Winrate&quot;, 
				(#winrateText &gt; 0) and table.concat(winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		:node( (isSuper) and p.s_row(
			&quot;Winrate&quot;, 
				(#s_winrateText &gt; 0) and table.concat(s_winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
	return tostring(infobox)
end

function p.referenceTest()
	local element = mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible&quot; )
		:css( &quot;text-align&quot;,&quot;left&quot; )

	local bigElement = mw.html.create( &quot;div&quot; )
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface&quot;)
		)
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface2&quot;)
		)

	return tostring(element) .. &quot;\n&quot; .. tostring(bigElement) .. &quot;\n&quot;
	
end

function p.roundedPct(num, den)
	return (num/den % 0.01 &lt; 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100)
end

function p.title()
	--[[local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;
		}
	}
	local cases2 = &quot;ba&quot;
	local cases3 = &quot;ba&quot;
	-- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = &quot;true&quot; else cases2 = &quot;false&quot; end
	-- if &quot;ANCOVA&quot; == frame:getParent():getTitle() then cases3 = &quot;true&quot; else cases3 = &quot;false&quot; end
	local detected = false
	detected = detected or true]]
	return mw.title.getCurrentTitle() 
end

function p.ifexist()
	return tonumber(frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot;, args = {frame.args[1]} })
end

function p.performance()
	for i = 1,5000000 do p.tableMake() end
	return
end

function p.performance1()
	local var = &quot;&quot;
	for i = 1,100000000 do if var[1] then end end
	return
end

function p.performance2()
	local var = &quot;&quot;
	for i = 1,100000000 do if type(var) == &quot;table&quot; then end end
	return
end

p.counter = 0
function p.tableBake()
	p.counter = p.counter + 1
	return p.counter
end


function p.tableMake()
	p.counter = 0
	local var = {
		keyword = {p.tableBake()},
		kljuczslovo = {p.tableBake()},
		yaoshiwenzi = {p.tableBake()},
		llavepalabra = {p.tableBake()},
	}
	return var
end

function p.expand()
	return frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot; }
end

function p.title3()
	return mw.title.getCurrentTitle().fullText
end

function p.getBallAttribute()
	p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4])
end

function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration )
	if _ then return mw.text.nowiki(ballData) 
	else return &quot;&quot; end
end

function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att]
	if ballData[1] then
		return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)]
	end
	return ballData
end

function p.emojiCodepointInternal( emoji )
	emoji = mw.text.trim( emoji )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.emojiCodepoint()
	local emoji = mw.text.trim( mw.getCurrentFrame().args[1] )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.title2()
	local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;,
			[&quot;Module:Sandbox/22&quot;] = &quot;Module:Sandbox/22&quot;
		}
	}
	
	local concepts = {
		grammatical_cases,
	}
	local PAGENAME = tostring(mw.title.getCurrentTitle())
	-- local detected = false
	local out = {}
	local debugger = {}
	for i, group in ipairs(concepts) do 
		if group.inessive[PAGENAME] then 
			table.insert(debugger,&quot;Okay, managed TRUE on &quot;..i..&quot;:&quot;..group.ablative)
			-- detected = detected or true
			local formattedGroup = {}
			for memberKey, memberRendervalue in pairs(group.inessive) do
				table.insert(formattedGroup, &quot;&lt;li&gt;&quot; .. memberRendervalue .. &quot;&lt;/li&gt;&quot;)
			end 
			table.insert(out, &quot;&lt;h3&gt;&quot; .. group.ablative .. &quot;&lt;/h3&gt;&lt;ul&gt;&quot; .. table.concat( formattedGroup ) .. &quot;&lt;/ul&gt;&quot; )
			table.insert(out, &quot;this sucks ass&quot;)
			return &quot;this sucks ass&quot;
		else
			table.insert(debugger,&quot;Managed FALSE on &quot;..i..&quot;:&quot;..group.ablative..&quot;. Comparator: &quot;..PAGENAME..&quot;, &quot;)
		end
	end
	return debugger
	--[[
	if detected then return tostring(detected) .. table.concat(out) end
	return
	--]]
end

function p.sanitizeChar(capture)
	-- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited
	if capture == &quot;?&quot; then
		return &quot;&quot;
	elseif capture == &quot;#&quot; then
		return &quot;No. &quot;
	end
end

function p.sanitizeTitle()
	local title = frame.args[1]
	return mw.ustring.gsub( title, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar) 
end

function p.sanitizeTitletest2()
	return mw.ustring.gsub( &quot;Ki vagy?  (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)&quot;, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar()) 
end

function p.safetyTest()
	if string.len(frame.args[1]) &lt; 500 then 
		return frame.args[1] 
	else 
		return &quot;&quot;
	end
end

--[[ 
function array_iter(t)
 	local i = 0
	return function ()
		i = i + 1
		return t[i] 
	end 
end
--]]

function p.listInline()
	--[[ This function is a godsend! 
	https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]]
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return mw.text.listToText( spritedTable , &quot;, &quot;, &quot;, and &quot; )
end

function p.listNavbox()
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return table.concat( spritedTable , &quot; • &quot;)
end

function p.listCountInternal(table) 
	local count = 0
	for index,value in ipairs( table ) do count = count + 1 end
	return count
end

function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn

function p.tableTest()
	return balls[1][1]
end

function p.todaysBall()
	--[[ https://www.lua.org/pil/3.6.html
		 The use of explicit indexing here is not strictly semantic;
		 it's just to emphasize the rotation order.
	 ]]
	local unixDay = math.floor(os.time()/86400)
	local ballTotalCount = 0
	local ballCounts = {}
    for whichTable,subtable in ipairs(balls) do 
		ballCounts[whichTable] = #subtable
		ballTotalCount = ballTotalCount + ballCounts[whichTable]
	end
	local cycleDayIndex,  cycleStartingBall = unixDay % ballTotalCount,  math.floor(unixDay/ballTotalCount) % ballTotalCount
	local cycleTodaysBall
	if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount
	elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount
	elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount
	else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end
	--[[ debug
	return cycleTodaysBall .. &quot; &quot; .. ballTotalCount .. &quot; &quot; .. os.time() .. &quot; &quot; .. math.floor(os.time()/86400) .. &quot; &quot; .. cycleDayIndex .. &quot; &quot; .. cycleStartingBall
	]]
	for whichTable = 1, #balls do 
		if ballCounts[whichTable] &gt; cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] 
		else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end
	end 
	--]]
end

return p;</text>
      <sha1>gxwxollrdv4irgbqm2d8pg9f0oq5bqx</sha1>
    </revision>
    <revision>
      <id>2834744</id>
      <parentid>2834743</parentid>
      <timestamp>2026-09-28T00:27:45Z</timestamp>
      <contributor>
        <username>Helpme2222</username>
        <id>3106525</id>
      </contributor>
      <origin>2834744</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="21074" sha1="askmv43c74dts626u97ws8dlh8b288a" xml:space="preserve">-- My function for teaching grammar

local p = {};
local frame = mw.getCurrentFrame()
local grammatical_moods = {
	&quot;Jelen ido&quot;,
	&quot;Mult ido&quot;,
	&quot;Felszolito mod&quot;,
	&quot;Felteteles mod&quot;
}
local balls = {
	grammatical_moods
}

local toggleButton = mw.html.create( 'span' )
	:css({
		float = &quot;right&quot;,
		[&quot;margin-right&quot;] = &quot;0&quot;,
		[&quot;font-weight&quot;] = &quot;normal&quot;
	})
	:addClass(&quot;mw-customtoggle-Infobox&quot;)
	:node(&quot;[Click to toggle]&quot;)

function p.subheading( discrim, b_content, content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node((discrim) and b_content or content)
			:node((discrim) and toggleButton)
		)
end
	
function p.s_subheading( s_content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible mw-collapsed&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node(s_content)
			:node(toggleButton)
		)
end

function p.row( discrim, name, data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; ) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(name)
		)
		:node( mw.html.create( 'td' )
			:node(data)
		)
end
	
function p.s_row( s_name, s_data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass(&quot;mw-collapsible mw-collapsed&quot;) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(s_name)
		)
		:node( mw.html.create( 'td' )
			:node(s_data)
		)
end

function p.var(symbol)
	return frame:expandTemplate{ title = &quot;Template:Var&quot;, args = {symbol} }
end
	

function p.infoboxWBB()
	-- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html
	--[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node 
			Thankfully, these operations interpret passed nils as no-ops 
			(I'm technically using false though. Somehow that works anyways?)]]
	
	local frame = mw.getCurrentFrame()
	local ballname = frame.args[&quot;Title&quot;] or tostring(mw.title.getCurrentTitle())
	local Entities = p -- require(&quot;Module:Entities&quot;)
	
	local featureCount = tonumber(frame.args[&quot;Feature count&quot;]) or 1
	local s_featureCount = tonumber(frame.args[&quot;SUPER Feature count&quot;]) or featureCount
	local matchCount = {
		total = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local matchWinCount = {
		standard = mw.site.stats.pagesInCategory( ballname..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local s_matchCount = {
		total = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local s_matchWinCount = {
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local isSuper = 
		frame.args[&quot;SUPER Feature count&quot;] or
		frame.args[&quot;SUPER Type&quot;] or 
		frame.args[&quot;SUPER Debuted&quot;] or
		(s_matchCount.total &gt; 0) or 
		nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra &quot;or nil&quot; had to be added since this is used in :addClass calls.
	-- TITLE
	local infobox = mw.html.create( 'table' )
		:addClass(&quot;wikitable&quot;)
		:css({
			float = &quot;right&quot;; 
			[&quot;margin-left&quot;] = &quot;10px&quot;
		})
		:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass((isSuper) and &quot;mw-collapsible&quot; or nil)
				:node(&quot;&lt;big&gt;&quot; .. ballname .. &quot;&lt;/big&gt;&quot;)
			)
		)
		:node((isSuper) and mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass(&quot;mw-collapsible mw-collapsed&quot;)
				:node(&quot;&lt;big&gt;&quot; .. (frame.args[&quot;SUPER Title&quot;] or (&quot;Super &quot; .. ballname)) .. &quot;&lt;/big&gt;&quot;)
			)
		)
	-- SECTION: Identification
		-- Section Heading
		:node( p.subheading( (frame.args[&quot;SUPER Type&quot;]), &quot;Base Identification&quot;, &quot;Basic Info&quot; ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_subheading( &quot;Super Identification&quot; ) )
		-- Type
		:node( p.row( (frame.args[&quot;SUPER Type&quot;]), &quot;Type&quot;, frame.args[&quot;Type&quot;] ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_row( &quot;Type&quot;, frame.args[&quot;SUPER Type&quot;] ) )
		-- Color
	local color = frame.args[&quot;Color&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;color&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Color&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(color ~= &quot;&quot;) and frame:expandTemplate{ title = &quot;Color&quot;, args = {color} } or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add color...&lt;/i&gt;]]&quot;
				)
			)
		)
		-- Emoji
	local emoji = frame.args[&quot;Emoji&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;emoji&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Emoji&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(emoji ~= &quot;&quot;) and ( emoji .. &quot; &lt;code&gt;&quot; .. p.emojiCodepointInternal(emoji) .. &quot;&lt;/code&gt;&quot;) or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add emoji...&lt;/i&gt;]]&quot;
				)
			)
		)
	-- SECTION: Scaling Characteristics
	--[[ Some data will be collected first. ]]
		-- (Base) Behavior
	local features = {}
	local starts = {}
	local scalings = {}
	if (featureCount &gt; 1) then
		for i = 1, featureCount do
			table.insert(features, 
				{
					frame.args[&quot;Feature &quot;..i], 
					&quot; (&quot; .. p.var(frame.args[&quot;Var &quot;..i]) .. &quot;)&quot;
				})
			table.insert(starts, 
				{
					p.var( frame.args[&quot;Var0 &quot;..i] or (frame.args[&quot;Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;, 
					frame.args[&quot;Starting value &quot;..i]
				})
			table.insert(scalings,
				{
					p.var( &quot;Δ&quot;..(frame.args[&quot;Var &quot;..i]) ) .. &quot;: &quot;,
					frame.args[&quot;Scaling &quot;..i]
				})
		end
	else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1
		features[1] = {frame.args[&quot;Feature&quot;],&quot;&quot;}
		starts[1] = {&quot;&quot;,frame.args[&quot;Starting value&quot;]}
		scalings[1] = {&quot;&quot;,frame.args[&quot;Scaling&quot;]}
	end
		-- Super Behavior
	local s_features
	local s_starts
	local s_scalings
	-- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default.
	local has_s_features;
	local has_s_starts;
	local has_s_scalings;
	if (frame.args[&quot;SUPER Feature count&quot;]) then
		if s_featureCount &gt; 1 then 
			-- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data
			s_features = mw.clone(features)
			s_starts = mw.clone(starts)
			s_scalings = mw.clone(scalings)
			-- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors.
			if s_featureCount &gt; featureCount then for i = featureCount + 1, s_featureCount do 
					table.insert( s_features, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_starts, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_scalings, {&quot;&quot;,&quot;&quot;} )
				end
			elseif s_featureCount &lt; featureCount then for i = s_featureCount + 1, featureCount do 
					s_features[i] = nil
					s_starts[i] = nil
					s_scalings[i] = nil
				end
			end
			-- Parse super data, set the has_s_... flags if triggered
			for i = 1,s_featureCount do
				if frame.args[&quot;SUPER Feature &quot;..i] then 
					s_features[i][1] = frame.args[&quot;SUPER Feature &quot;..i] 
					if (not has_s_features) then has_s_features = true end
				end
				if frame.args[&quot;SUPER Starting value &quot;..i] then 
					s_starts[i][2] = frame.args[&quot;SUPER Starting value &quot;..i] 
					if (not has_s_starts) then has_s_starts = true end
				end
				if frame.args[&quot;SUPER Scaling &quot;..i] then 
					s_scalings[i][2] = frame.args[&quot;SUPER Scaling &quot;..i] 
					if (not has_s_scalings) then has_s_scalings = true end
				end
				if frame.args[&quot;SUPER Var &quot;..i] then
					s_features[i][2] = &quot; (&quot; .. p.var(frame.args[&quot;SUPER Var &quot;..i]) .. &quot;)&quot;
					s_starts[i][1] = p.var( frame.args[&quot;SUPER Var0 &quot;..i] or (frame.args[&quot;SUPER Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;
					s_scalings[i][1] = p.var( '&lt;span style=&quot;font-style:normal&quot;&gt;Δ&lt;/span&gt;'..(frame.args[&quot;SUPER Var &quot;..i]) ) .. &quot;: &quot;
					if (not has_s_features) then has_s_features = true end
					if (not has_s_starts) then has_s_starts = true end
					if (not has_s_scalings) then has_s_scalings = true end
				end 
				s_features[i] = table.concat(s_features[i])
				s_starts[i] = table.concat(s_starts[i])
				s_scalings[i] = table.concat(s_scalings[i])
			end
			s_features = table.concat(s_features,&quot;&lt;br&gt;&quot;)
			s_starts = table.concat(s_starts,&quot;&lt;br&gt;&quot;)
			s_scalings = table.concat(s_scalings,&quot;&lt;br&gt;&quot;)
		else -- # of super features = 1? just write directly to the variable
			if frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Feature&quot;] then 
				has_s_features = true
				s_features = frame.args[&quot;SUPER Feature&quot;]
			end
			if frame.args[&quot;SUPER Var0&quot;] or frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Starting value&quot;] then 
				has_s_starts = true
				s_starts = frame.args[&quot;SUPER Starting value&quot;]
			end
			if frame.args[&quot;SUPER Scaling&quot;] then 
				has_s_scalings = true
				s_scalings = frame.args[&quot;SUPER Scaling&quot;]
			end
		end
	end
		-- finalize base data as strings now that work is otherwise complete
	if (featureCount &gt; 1) then 
		for i = 1, featureCount do
			features[i] = table.concat(features[i])
			starts[i] = table.concat(starts[i])
			scalings[i] = table.concat(scalings[i])
		end 
		features = table.concat(features,&quot;&lt;br&gt;&quot;)
		starts = table.concat(starts,&quot;&lt;br&gt;&quot;)
		scalings = table.concat(scalings,&quot;&lt;br&gt;&quot;)
	else
		features = features[1][1]
		starts = starts[1][2]
		scalings = scalings[1][2]
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (frame.args[&quot;SUPER Feature count&quot;]), &quot;Base Scaling Characteristics&quot;, &quot;Scaling Characteristics&quot; ) )
		:node( (frame.args[&quot;SUPER Feature count&quot;]) and p.s_subheading( &quot;Super Scaling Characteristics&quot; ) )
		-- Features
		:node( p.row( (has_s_features), 
				(featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			features
		) )
		:node( (has_s_features) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;SUPER Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;SUPER Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			s_features
		) )
		-- Starting values
		:node( p.row( (has_s_starts), 
				(featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;Var0&quot;] or (frame.args[&quot;Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			starts
		) )
		:node( (has_s_starts) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;SUPER Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;SUPER Var0&quot;] or (frame.args[&quot;SUPER Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			s_starts
		) )
		-- Scaling
		:node( p.row( (has_s_scalings), &quot;Scaling&quot;, scalings) )
		:node( (has_s_scalings) and p.s_row(&quot;Scaling&quot;, s_scalings) )
	-- SECTION: Career 
	--[[ Some data will be collected first. ]]
		-- (Base) Games played
	local gamesPlayedText = {}
	if matchCount.total &gt; 0 then
		table.insert(gamesPlayedText, matchCount.total .. &quot; (total)&quot;)
		if matchCount.standard &gt; 0 then table.insert(gamesPlayedText, matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- (Base) Winrate
	local winrateText = {}
	if matchCount.total &gt; 0 then
		if matchCount.standard &gt; 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
		-- Super Games played
	local s_gamesPlayedText = {}
	if s_matchCount.total &gt; 0 then
		table.insert(s_gamesPlayedText, s_matchCount.total .. &quot; (total)&quot;)
		if s_matchCount.standard &gt; 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- Super Winrate
	local s_winrateText = {}
	if s_matchCount.total &gt; 0 then
		if s_matchCount.standard &gt; 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (isSuper), &quot;Base Career&quot;, &quot;Career &quot;) )
		:node( (isSuper) and p.s_subheading( &quot;Super Career&quot;) )
		-- Debuted
		:node( p.row( (isSuper), &quot;Debut&quot;, frame.args[&quot;Debut&quot;] ) )
		:node( (isSuper) and p.s_row(&quot;Debut&quot;, frame.args[&quot;SUPER Debut&quot;]) )
		-- Games played
		:node( p.row( 
			(isSuper), 
			&quot;Games played&quot;, 
				(#gamesPlayedText &gt; 0) and table.concat(gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot; 
		) )
		:node( (isSuper) and p.s_row(
			&quot;Games played&quot;, 
				(#s_gamesPlayedText &gt; 0) and table.concat(s_gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		-- Winrate
		:node( p.row( 
			(isSuper), 
			&quot;Winrate&quot;, 
				(#winrateText &gt; 0) and table.concat(winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		:node( (isSuper) and p.s_row(
			&quot;Winrate&quot;, 
				(#s_winrateText &gt; 0) and table.concat(s_winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
	return tostring(infobox)
end

function p.referenceTest()
	local element = mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible&quot; )
		:css( &quot;text-align&quot;,&quot;left&quot; )

	local bigElement = mw.html.create( &quot;div&quot; )
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface&quot;)
		)
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface2&quot;)
		)

	return tostring(element) .. &quot;\n&quot; .. tostring(bigElement) .. &quot;\n&quot;
	
end

function p.roundedPct(num, den)
	return (num/den % 0.01 &lt; 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100)
end

function p.title()
	--[[local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;
		}
	}
	local cases2 = &quot;ba&quot;
	local cases3 = &quot;ba&quot;
	-- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = &quot;true&quot; else cases2 = &quot;false&quot; end
	-- if &quot;ANCOVA&quot; == frame:getParent():getTitle() then cases3 = &quot;true&quot; else cases3 = &quot;false&quot; end
	local detected = false
	detected = detected or true]]
	return mw.title.getCurrentTitle() 
end

function p.ifexist()
	return tonumber(frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot;, args = {frame.args[1]} })
end

function p.performance()
	for i = 1,5000000 do p.tableMake() end
	return
end

function p.performance1()
	local var = &quot;&quot;
	for i = 1,100000000 do if var[1] then end end
	return
end

function p.performance2()
	local var = &quot;&quot;
	for i = 1,100000000 do if type(var) == &quot;table&quot; then end end
	return
end

p.counter = 0
function p.tableBake()
	p.counter = p.counter + 1
	return p.counter
end


function p.tableMake()
	p.counter = 0
	local var = {
		keyword = {p.tableBake()},
		kljuczslovo = {p.tableBake()},
		yaoshiwenzi = {p.tableBake()},
		llavepalabra = {p.tableBake()},
	}
	return var
end

function p.expand()
	return frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot; }
end

function p.title3()
	return mw.title.getCurrentTitle().fullText
end

function p.getBallAttribute()
	return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4])
end

function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration )
	if _ then return mw.text.nowiki(ballData) 
	else return &quot;&quot; end
end

function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att]
	if ballData[1] then
		return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)]
	end
	return ballData
end

function p.emojiCodepointInternal( emoji )
	emoji = mw.text.trim( emoji )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.emojiCodepoint()
	local emoji = mw.text.trim( mw.getCurrentFrame().args[1] )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.title2()
	local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;,
			[&quot;Module:Sandbox/22&quot;] = &quot;Module:Sandbox/22&quot;
		}
	}
	
	local concepts = {
		grammatical_cases,
	}
	local PAGENAME = tostring(mw.title.getCurrentTitle())
	-- local detected = false
	local out = {}
	local debugger = {}
	for i, group in ipairs(concepts) do 
		if group.inessive[PAGENAME] then 
			table.insert(debugger,&quot;Okay, managed TRUE on &quot;..i..&quot;:&quot;..group.ablative)
			-- detected = detected or true
			local formattedGroup = {}
			for memberKey, memberRendervalue in pairs(group.inessive) do
				table.insert(formattedGroup, &quot;&lt;li&gt;&quot; .. memberRendervalue .. &quot;&lt;/li&gt;&quot;)
			end 
			table.insert(out, &quot;&lt;h3&gt;&quot; .. group.ablative .. &quot;&lt;/h3&gt;&lt;ul&gt;&quot; .. table.concat( formattedGroup ) .. &quot;&lt;/ul&gt;&quot; )
			table.insert(out, &quot;this sucks ass&quot;)
			return &quot;this sucks ass&quot;
		else
			table.insert(debugger,&quot;Managed FALSE on &quot;..i..&quot;:&quot;..group.ablative..&quot;. Comparator: &quot;..PAGENAME..&quot;, &quot;)
		end
	end
	return debugger
	--[[
	if detected then return tostring(detected) .. table.concat(out) end
	return
	--]]
end

function p.sanitizeChar(capture)
	-- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited
	if capture == &quot;?&quot; then
		return &quot;&quot;
	elseif capture == &quot;#&quot; then
		return &quot;No. &quot;
	end
end

function p.sanitizeTitle()
	local title = frame.args[1]
	return mw.ustring.gsub( title, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar) 
end

function p.sanitizeTitletest2()
	return mw.ustring.gsub( &quot;Ki vagy?  (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)&quot;, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar()) 
end

function p.safetyTest()
	if string.len(frame.args[1]) &lt; 500 then 
		return frame.args[1] 
	else 
		return &quot;&quot;
	end
end

--[[ 
function array_iter(t)
 	local i = 0
	return function ()
		i = i + 1
		return t[i] 
	end 
end
--]]

function p.listInline()
	--[[ This function is a godsend! 
	https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]]
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return mw.text.listToText( spritedTable , &quot;, &quot;, &quot;, and &quot; )
end

function p.listNavbox()
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return table.concat( spritedTable , &quot; • &quot;)
end

function p.listCountInternal(table) 
	local count = 0
	for index,value in ipairs( table ) do count = count + 1 end
	return count
end

function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn

function p.tableTest()
	return balls[1][1]
end

function p.todaysBall()
	--[[ https://www.lua.org/pil/3.6.html
		 The use of explicit indexing here is not strictly semantic;
		 it's just to emphasize the rotation order.
	 ]]
	local unixDay = math.floor(os.time()/86400)
	local ballTotalCount = 0
	local ballCounts = {}
    for whichTable,subtable in ipairs(balls) do 
		ballCounts[whichTable] = #subtable
		ballTotalCount = ballTotalCount + ballCounts[whichTable]
	end
	local cycleDayIndex,  cycleStartingBall = unixDay % ballTotalCount,  math.floor(unixDay/ballTotalCount) % ballTotalCount
	local cycleTodaysBall
	if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount
	elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount
	elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount
	else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end
	--[[ debug
	return cycleTodaysBall .. &quot; &quot; .. ballTotalCount .. &quot; &quot; .. os.time() .. &quot; &quot; .. math.floor(os.time()/86400) .. &quot; &quot; .. cycleDayIndex .. &quot; &quot; .. cycleStartingBall
	]]
	for whichTable = 1, #balls do 
		if ballCounts[whichTable] &gt; cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] 
		else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end
	end 
	--]]
end

return p;</text>
      <sha1>askmv43c74dts626u97ws8dlh8b288a</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment</title>
    <ns>0</ns>
    <id>331021</id>
    <revision>
      <id>2834783</id>
      <parentid>2834661</parentid>
      <timestamp>2026-09-28T03:50:49Z</timestamp>
      <contributor>
        <username>StretchBeyond</username>
        <id>3105744</id>
      </contributor>
      <comment>/* Why treatment can be difficult */ Improved a poorly worded paragraph as suggested by a Wiki user, and also removed footnote references, replacing them with standard Wiki hyperlinks.</comment>
      <origin>2834783</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35073" sha1="i0zsk9q2k5j5rakfz7y02hpb6l0wqi6" xml:space="preserve">{{title|Immersive therapy for PTSD treatment:&lt;br&gt;How does it work and what are the effects?}}
__TOC__
== Overview ==
{{RoundBoxTop|theme=2}}
'''Scenario: Immersive PTSD therapies'''

Andriy, a soldier, harnessed to a treadmill, walks towards an image he spent months avoiding, his clinical therapist  beside him. This is multi-modular motion assisted memory desensitisation and reconsolidation (3MDR), one of a new generation of immersive therapies for post-traumatic stress disorder. 
[[File:Andriy 3MDR enhanced.gif|center|600x600px]]

'''Figure 1.''' Andriy moves through his 3MDR treatment.

Learn about immersive therapy and Andriy’s* experience below (*Andriy: a fictional name). {{RoundBoxBottom}}
[[File:Post-traumatic_stress_disorder_world_map_-_DALY_-_WHO2004.svg|alt=|thumb|271x271px|'''Figure 2:''' The World Health Organisation (2024) estimated that [https://www.who.int/news-room/fact-sheets/detail/post-traumatic-stress-disorder 3.9% of the world's population had PTSD] at some stage.]]
[[w:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (PTSD) develops after severe or life-threatening trauma and carries a substantial personal and societal cost (Figure 2), with military personnel disproportionately represented (Boska et al., 2025). See costs and limitations below (Davis et al., 2022; Montgomery-Marks et al., 2025). 

PTSD's emotional impact is shaped by [[w:Emotional_dysregulation|emotional dysregulation]] - difficulty managing intense feelings such as guilt, fear or shame (Westphal et al., 2017). This commonly triggers [[Cognitive psychology|cognitive]] and behavioural avoidance that offers short-term relief but prevents traumatic memory from being adaptively processed, trapping individuals in cycles of avoidance and chronic hyperarousal (de Haart et al., 2026; van Gelderen et al., 2018).

Immersive interventions, including [[w:Virtual_reality_therapy|virtual reality exposure therapy]] (VRET) and 3MDR, aim to break this avoidance cycle by creating controlled environments in which trauma cues can be safely approached rather than avoided (van Gelderen et al., 2018; Wiederhold &amp; Wiederhold, 2025). This chapter explains the psychological theory behind these approaches, reviews the research evidence for their effects and considers their limitations.

{{RoundBoxTop|theme=2}}
'''Focus questions'''[[File:Crystal Clear app ktip.svg|left|20px|]]

* Why is emotional processing important in PTSD?
* How can immersive therapies influence the emotional processes underlying PTSD? 
* What does the research evidence show?
* What are the costs and limitations of immersive therapies? 
{{RoundBoxBottom}}

== Why is emotional processing important in PTSD? ==
[[File:PTSD.png|thumb|'''Figure 3.''' PTSD can deeply affect our emotions.]]PTSD is formally diagnosed according to [[w:DSM-5|DSM-5]] criteria and is characterised by disrupted executive and emotional processing systems, heightened threat perception, hypervigilance and persistent negative emotional states as depicted in Figure 3 (Kukharuk et al., 2025; López-Ojeda &amp; Hurley, 2022; Osman et al., 2016). Emotional processing theory provides the primary framework for understanding this disruption, supported by emotion regulation theory and inhibitory learning models introduced later in the chapter.

{{RoundBoxTop|theme=3}}[[Image:Crystal Clear app help index.svg|left|50px]]
;Predict the outcome

A soldier with PTSD encounters a trauma-related image expecting:

'''TRAUMA CUE → DANGER → DISTRESS → AVOID'''

However, during immersive treatment, the expected danger does not occur. What is the most likely consequence?&lt;quiz display=simple&gt;
{
|type=&quot;()&quot;}
- Fear increases permanently.

- Memory cannot change.

+ Prediction error creates opportunities for new learning.

- Emotional processing stops. 

}

&lt;/quiz&gt;
&lt;div style=&quot;text-align:right; color:red; font-weight:bold;&quot;&gt;
Click &quot;show&quot; below  to understand more⤵ &lt;/div&gt;
{{Hidden begin|title=Please consider the answer before opening this section}}The answer is '''C'''.
&lt;div style=&quot;text-align:left; color:black; font-weight:regular;&quot;&gt;
Mismatches between expected and actual danger create a
'''prediction error'''. This may contribute to fear extinction,
emotion regulation, and memory reconsolidation. '''Think about it:''' If approaching rather than avoiding trauma can create new learning, what role might emotion regulation, fear extinction, prediction error and memory reconsolidation play?
&lt;/div&gt;

 
{{Hidden end}}
&lt;div style=&quot;text-align:left; color:black; font-weight:regular;&quot;&gt;
&lt;/div&gt;{{RoundBoxBottom}}

=== Understanding PTSD and emotions ===
Emotional processing theory suggests recovery requires the trauma memory to be reactivated and updated with corrective information. In PTSD, this process is blocked and to manage distress, many individuals adopt cognitive and behavioural avoidance mechanisms (López-Ojeda &amp; Hurley, 2022). Avoidance offers short-term relief, but prevents the traumatic memory from being reactivated, so it cannot be updated. Trauma reminders such as flashbacks continue to trigger extreme distress, trapping the individual in a maladaptive, self-reinforcing avoidance cycle (Figure 4)  (Vermetten, Burback, et al., 2025b). 

At a neural level, Westphal et al. (2017) linked this to [[wikipedia:Transdiagnostic_process|transdiagnostic]] [[w:Emotional_dysregulation|emotion dysregulation]], in which the traumatic memory network remains isolated from the brain's [[w:Salience_network|salience]]  and central executive networks. Effective treatment requires reactivating this network so the memory can be integrated (Vermetten, Burback, et al., 2025b; Westphal et al., 2017). This is empirically supported: A 2025 study of Danish military veterans (''n''=142) found emotional regulation difficulties explained an additional 28% of the variance in PTSD symptoms; combined with [[w:Comorbidity|comorbid]] symptoms, these factors accounted for 52% of the variance in severity (''F''(13, 92) = 9.58, ''p'' &lt;0.001) (Elklit &amp; Dahl, 2025).[[File:Avoidance and Processing Cycles.png|500x500px|thumb|'''Figure 4'''. Avoidance cycles and the anticipated positive responses associated with immersive therapies. Based on: Lopez-Ojeda et al. (2022), Felemban et al. (2026), and Vermetten et. al. (2025b).|center]]

=== Why treatment can be difficult ===
Trauma-focused psychotherapies such as [[w:Prolonged_exposure_therapy|prolonged exposure]] (PE) and [[w:Cognitive_processing_therapy|cognitive processing therapy]] (CPT) ask patients to actively engage with distressing memories to generate fear extinction, precisely what avoidance prevents (van Toorenburg et al., 2020). Many patients cannot tolerate the emotional exposure these therapies require (Lopes et al., 2025; van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). Because the trigger is never confronted, the brain cannot experience a prediction error needed to learn the threat has passed, so symptoms persist (de Haart et al., 2026; López-Ojeda &amp; Hurley, 2022).

This is reflected in outcomes where an estimated 39.2% of patients fail to respond to standard trauma-focused therapy, and dropout rates range from 16-48% (de Haart et al., 2026; Vermetten, Burback, et al., 2025b). Among military veterans, younger patients show heightened severity when trauma is central to their identity, while others turn to poor diet or substance abuse, further eroding emotional regulation (Efremov, 2025; Niles et al., 2023).

[[w:Emotional_dysregulation|Emotional dysregulation]] was historically viewed as a fixed barrier requiring lengthy stabilisation before treatment could begin (van Toorenburg et al., 2020). More recent evidence suggests otherwise, although emotional regulation contributes to PTSD symptoms, it is also being increasingly viewed as a modifiable process with dynamic capacity, one able to improve as a natural consequence of successful memory processing (van Toorenburg et al., 2020). This reframes the clinical challenge as helping an individual safely approach and process the traumatic memory, not correcting a fixed deficit. [[File:UC PhD VR study.png|right|thumb|'''Figure 5.''' PhD work by Selvakumaran (University of Canberra) examining cultural adaptation using the US-''Bravemind'' system.]]Cultural context adds another layer of difficulty. Doctoral research at the [[University of Canberra]] is integrating exposure therapy with gamified, posture-adaptive cognitive rehabilitation to support physical and emotional recovery (Figure 5). Selvakumaran (2025), is examining how existing protocols such as used in the US-centric ''[https://medvr.ict.usc.edu/projects/bravemind.html Bravemind]'' system need cultural adaptation for Australian veterans and first responders, because their operational context and backgrounds differ from their US counterparts (Selvakumaran, 2025). These treatment gaps carry a substantial economic and personal cost, part of the motivation for developing alternatives such as immersive therapies.   

== How can immersive therapies influence emotional processes? ==
Immersive PTSD treatments mark a shift from passive, sedentary therapy towards active, embodied, interactive approaches (van Gelderen et al., 2018). Immersive therapy uses three psychological mechanisms and their interaction: multisensory presence, embodied cognition, and divergent thinking (López-Ojeda &amp; Hurley, 2022; van Gelderen et al., 2018).

Embodied cognition is how physical states of the body can directly modify states of the mind (van Gelderen et al., 2018). Immersion therapy goes beyond simple visual replication of a trauma memory; instead, capturing the participant’s visceral and cognitive focus by limiting distractions (Macey et al., 2026). 

=== What is immersive therapy? ===
[[w:Immersion_therapy|Immersive therapies]] use [[w:Extended_reality|extended reality]] (XR) platforms, encompassing [[virtual reality]] (VR), [[Augmented Reality|augmented reality]] (AR), and [[w:Mixed_reality_game|mixed reality]] (MR), to create customisable, controlled, and standardised therapeutic environments (López-Ojeda &amp; Hurley, 2022; Wiederhold &amp; Wiederhold, 2025). This ecosystem is depicted in Figure 6.
This chapter focuses on two applications: VRET where therapists reconstruct traumatic scenarios in safe, graded environments, and 3MDR, which extends this by having the patient move on a treadmill beside their therapist, rather than in a stationary, face-to-face session (de Haart et al., 2026; Felemban et al., 2026).[[File:XR and Human Senses.png|thumb|'''Figure 6.''' The Extended Reality environment. |center|750x750px]]
=== Presence and embodied cognition ===
Immersive therapy’s distinguishing feature is its ability to generate presence, or the psychological illusion of being ‘''there’''. The illusion is amplified by integrating synchronised audio, visual, [[w:Olfactory|olfactory]], [[wikipedia:Haptic_perception|haptic]], and movement stimuli (Lopes et al., 2025; López-Ojeda &amp; Hurley, 2022). This increases engagement with the trauma memory and supports emotional processing (van Gelderen et al., 2018).

In 3MDR, presence combines with cognition. The working principle is that physical states of the body can directly shape states of the mind (van Gelderen et al., 2018). Walking towards a virtual trauma reminder functions as a fear antagonistic action and rather than retreating in avoidance, the patient approaches, converting passive helplessness into active, empowered participation and disrupting the rigid and repetitive trauma narratives  common in PTSD (de Haart et al., 2026; Osman et al., 2016; van Gelderen et al., 2018). 

Notably, extinction learning normally requires moderate-to-high-intensity activity to stimulate [[w:Brain-derived_neurotrophic_factor|brain-derived neurotrophic factor]] (BDNF), but 3MDR's walking pace (&lt; 4 km/h) is too slow to generate meaningful BDNF secretion, making a major exercise induced BDNF mechanism unlikely, although physiological factors cannot be ruled out. The approach action and the cognitive restructuring it enables may be more important psychological drivers (de Haart et al., 2026).

=== Prediction error and inhibitory learning ===
Immersive therapy builds on the inhibitory learning model of exposure therapy, in which a new, safe association competes with and suppresses the original fear response. Walking towards a trauma cue and encountering safety instead of the expected catastrophe creates a prediction error between the anticipated, life-threatening event and actual reality. This may destabilise the traumatic memory, potentially creating conditions for memory reconsolidation to occur (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b; Wiederhold &amp; Wiederhold, 2025).

=== Memory reconsolidation ===
According to [[w:Memory_consolidation|memory reconsolidation]] theory, traumatic memories retrieved in a safe immersive context can become [[w:Malleability_of_intelligence|malleable]], allowing new, safe information to reconsolidate the memory in a non-threatening form (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). To prevent the patient from becoming overwhelmed, 3MDR uses dual-attention tasks such as tracking an oscillating ball (Figure 1), which tax working memory resources and reduce the vividness and emotional intensity of the memory (Vermetten, Burback, et al., 2025b).

Emerging linguistic research suggests this processing is reflected in patients' language. Across successive 3MDR sessions, [[w:Affect_labeling|affective labelling]] of feelings such as guilt shifted from past-tense to present-tense narration, consistent with a renewed ability to integrate traumatic memories into present-moment awareness (Vermetten, Barcaro, et al., 2025a).

== What does the research evidence show? ==
Addressing these barriers is important for any PTSD population, but especially critical for military populations with high treatment failure and dropout rates (van Gelderen et al., 2018). Immersive therapies can tailor patient-selected trauma cues to improve access to traumatic memory networks (Vermetten, Burback, et al., 2025b). 

=== Virtual reality exposure therapy ===
[[w:Virtual_reality_therapy|VRET]] reconstructs traumatic events in a structured context (Felemban et al., 2026; López-Ojeda &amp; Hurley, 2022). This may help bypass imagination challenges such as emotional numbing or amnesia that can prevent patients from engaging with traditional therapy (Macey et al., 2026). A meta-analysis of VRET for PTSD found substantial symptom reductions, averaging a 33.73-point decrease in the 0-80 point [[w:Clinician_Administered_PTSD_Scale|Clinician-Administered PTSD Scale]] and a 20.96-point decrease in the 0-80 point [https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PTSD Checklist PCL-5) scale]  (Felemban et al., 2026). Because changes of 10-20 points on these scales are considered clinically significant, this could mean the difference between severe functional impairment and mild or subclinical symptoms (Boska et al., 2025; de Haart et al., 2026; Felemban et al., 2026). Comparative effects against other active PTSD treatments remain modest, but VRET appears to be a more engaging alternative to conventional treatment (Felemban et al., 2026).

=== 3MDR ===
3MDR takes the same multisensory effect in VRET and adds an activating context. Rather than a sedentary, face-to-face session, the patient and the therapist face the virtual display together (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). It follows a three-phase protocol: pre-platform preparation, platform treadmill exposure, and post-platform re-consolidation (Vermetten, Burback, et al., 2025b). In a trial involving treatment-resistant PTSD; 3MDR showed large effect sizes from pre-treatment to six-months (''n'' = 134, ''d'' = 1.0) and high-acceptability, with 7-20% dropout rates (de Haart et al., 2026; van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). Substantially lower than the 16-48% typical trauma-focused therapy in military populations (Lewis et al., 2020).

Improvements are not limited to PTSD symptoms. A trial of 62 adults with severe PTSD, including childhood sexual trauma, found intensive trauma-focused treatment improved emotion-regulation abilities regardless of PTSD outcome (van Toorenburg et al., 2020). Some researchers link this improvement to positive psychology [[w:Broaden-and-build|broaden-and-build theory]]. Theorising that as patients regain a sense of safety and control, this may support a positive spiral of emotional flexibility that reinforces recovery (Fredrickson, 2001; Niles et al., 2023; Westphal et al., 2017).  

Critically, the exact causation remains unknown. Given the recent innovations in immersive therapies and small study samples, it remains unclear whether they work through exposure-based extinction, enhanced emotional engagement, physiological factors, embodiment or positive cognitive restructuring. These questions are relevant to both basic and applied research and have not yet been adequately addressed by current trial designs.  

'''Table 1'''

''Treatment Effects and Psychological Mechanisms''
{| class=&quot;wikitable mw-collapsible&quot;
| valign=&quot;top&quot; |'''Clinical Dimension'''
| valign=&quot;top&quot; |'''Traditional'''
| valign=&quot;top&quot; |'''Immersive'''
| valign=&quot;top&quot; |'''Psychological Mechanisms'''
|-
| valign=&quot;top&quot; |'''Context'''
| valign=&quot;top&quot; |'''''Sedentary'''''. Face-to-face, verbally describes trauma.
| valign=&quot;top&quot; |'''''Activating.''''' Dynamic, multi-sensory environment.
| valign=&quot;top&quot; |'''''Fear Antagonistic Action.'''''
'''''Approach behaviours.'''''

'''''Prediction Errors.'''''
|-
| valign=&quot;top&quot; |'''Trauma cue delivery'''
| valign=&quot;top&quot; |'''''Imaginary Retrieval'''''
Patient capacity
| valign=&quot;top&quot; |'''''Multisensory Immersion'''''
Highly tailored
| valign=&quot;top&quot; |'''''External Scaffolding.'''''
Bypasses internal barriers to activate memory networks.
|-
| valign=&quot;top&quot; |'''Processing'''
| valign=&quot;top&quot; |'''''Convergent processing.'''''
Repeated narration and fear habituation.
| valign=&quot;top&quot; |'''''Active Narrative Processing.'''''
Interactive, real-time affective labelling and dual attention tasks.
| valign=&quot;top&quot; |'''''Working Memory.'''''
Memory taxation reduces vividness and emotional intensity.
|-
| valign=&quot;top&quot; |'''Engagement''' 
| valign=&quot;top&quot; |'''''Attrition.'''''
Dropout rates 16-48%.
| valign=&quot;top&quot; |'''''Acceptability.'''''
Attractive. Dropout rates 7-20%.
| valign=&quot;top&quot; |'''''Sustained Motivation.'''''
Presence and safety in immersive environment.
|}
=== Applied example: 3MDR treatment in Ukraine ===
{{RoundBoxTop|theme=2}}
'''Scenario: Ukraine War: an applied 3MDR example'''

Andriy*, a Ukrainian Armed Forces volunteer, is struggling with hyper-vigilance, anxiety, and depression (Kukharuk et al., 2025). Traditional ‘talk’ therapy feels impossible; his mind remains in a constant state of combat readiness, even in quiet rooms.

In the pre-platform phase, Andriy identifies ‘hotspot’ memories in photographs. On the treadmill, harnessed and walking, he faces a panoramic screen as personalised warm-up music plays, selected to keep him connected to his traumatic memory network (Vermetten et al., 2025b). As his hotspot image fills the screen, his therapist asks three questions (Vermetten et al., 2025b):

1. What do you '''SEE''' ?
2. What does it '''TELL''' you?
3. What do you '''FEEL''' in your body NOW?

When Andriy identifies a surge of shame, the word '''GUILT''' is displayed as an affective label, and a dual-attention task begins. He tracks an oscillating, numbered ball while he stays with the emotion, taxing his working memory and reducing the intensity of the recalled trauma.
[[File:Andriy 3MDR enhanced.gif|center|600x600px]]

'''Figure 7.''' Andriy's 3MDR treatment.

By the end of the session, Andriy had walked toward what he used to avoid, creating a mismatch between his expectation of threat and his safety and turning a rigid, stuck memory into a manageable narrative.

*''Andriy, a fictional name given to one of 69 Ukrainian veterans, participating in a 2023 randomised controlled trial. Many demobilised after one year because of mental or neurological injuries'' (Kukharuk et al., 2025). 

{{RoundBoxBottom}}

== What are the costs and limitations of immersive therapies? ==
PTSD carries a substantial economic burden. An estimated US$232 billion in excess costs in the United States in 2018 and over £40 billion, in the United Kingdom in 2020-21, 92.4% of which were indirect rather than direct clinical costs (Davis et al., 2022; Montgomery-Marks et al., 2025). In Australia, the annual cost of PTSD per military veteran was estimated at $112,172 in 2025 (Magnusson &amp; Dey, 2025). In addition, individual healthcare costs surge by 142% in the year following a PTSD diagnosis, with comorbidities tripling this effect (Bothe et al., 2020). These figures span several countries and years and should be read as an indication of scale rather than as directly comparable totals.

Immersive therapy’s ability to move from small trials into mainstream PTSD treatment remains limited by methodological [[w:Homogeneity_and_heterogeneity|heterogeneity]], small sample sizes, and a lack of long-term data (Felemban et al., 2026). Many trials rely on military-heavy, treatment-resistant samples with few active comparison conditions, which limits the generalisation of findings to civilian or initial treatment populations. Practical issues such as [[w:Virtual_reality_sickness|cyber sickness]] or motion sickness can disrupt participation (Kukharuk et al., 2025).

Structural barriers include workforce training and equipment costs. Basic VR systems cost around US$3,500 per provider headset annually, with advanced simulation environments costing up to US$200,000 (Garrett et al., 2018). Despite this, adoption of immersive therapy is scaling. The [[w:United_States_Department_of_Veterans_Affairs|United States Veterans Affairs]] have expanded VR use from five medical centres in 2017 to over 154 centres and 2,300 staff, with applications now including over 40 documented clinical interventions such as chronic pain and suicide intervention (Bailey et al., 2024). Reflecting this potential growth trend, some market analysts estimate the global PTSD-focused VR therapy market was worth US$1.59 billion in 2025, forecasted to reach US$5.94 billion by 2032, driven largely by growing mental health awareness and the absence of standard clinical protocols (Stratistics MRC, 2025, an industry market report).

These limitations do not undermine the case for immersive therapy, but they show its evidence base and infrastructure are still maturing. Demonstrating rigorously why these therapies work, rather than assuming their novelty accounts for their effects is essential to avoid misallocating resources to unproven interventions and may support an emerging shift towards greater emphasis on post-traumatic growth and renewed optimism (Trejo et al., 2015; Vermetten, Burback, et al., 2025b; Wiederhold &amp; Wiederhold, 2025).

{{Robelbox|theme=2|title=Quiz}}
&lt;quiz display=simple&gt;
{Which of the following is an expected outcome of immersive therapy ?
|type=&quot;()&quot;}


- Patients are at risk because of an uncontrolled environment.
- The environments represent traditional face-to-face treatments.
- Failure and dropout rates are higher than traditional PTSD treatment.
+ Patients often broaden and build an upward spiral of emotion and optimism.
&lt;/quiz&gt;
{{Robelbox/close}}
== Conclusion ==
PTSD traps people in a self-reinforcing cycle of avoidance that blocks the adaptive processing necessary for traumatic memories to be resolved (see: ''[[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#Understanding PTSD and emotions|Understanding PTSD and emotions]]''). Standard trauma-focused therapies require patients to confront exactly what this cycle causes them to avoid, which contributes to high non-response and dropout rates (see ''[[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#Why treatment can be difficult|Why treatment can be difficult]]'').

Immersive therapies such as VRET and 3MDR address this by using presence, embodied cognition, prediction error and memory reconsolidation to help patients safely approach trauma cues rather than avoid them (see: ''[[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#What does the research evidence show?|What does the research evidence show]]''). The technology itself is only a delivery mechanism that enables the processes, the positive benefits appear to arise from the psychological mechanisms that immersive therapies facilitate, rather than the novelty of technology alone.

The evidence to date shows meaningful reductions in PTSD symptoms and comparatively low drop-out rates, alongside broader gains in emotional regulation. However, small samples, methodological variation, and a lack of long-term data mean the evidence base is still developing, and cost and infrastructure barriers remain significant (see: [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#What are the costs and limitations of immersive therapies?|costs and limitations]]'').{{RoundBoxTop|theme=11}}
[[File:Nuvola apps kuser.svg|Nuvola apps kuser|left|20px]] ''' Take-home message:''' 
Immersive therapies do not heal PTSD trauma through technological novelty or digital feedback. Instead, they create a safe, dynamic space that lets individuals confront trauma and reprocess memories into something they can live with. In doing so, people’s lives may once again shift toward value, optimism, and wellness.{{RoundBoxBottom}}

== See also ==
'''Wikiversity'''
* [[Motivation and emotion/Textbook/Emotion/Anxiety|Anxiety]] (Book chapter, 2010)
* [[/Non-invasive brain stimulation techniques/]] (Sub-page)
* [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)|Evidence-based PTSD assessment]]
* [[Motivation and emotion/Book/2024/Sense hacking|Sense hacking]] (Book chapter, 2024)

'''Wikipedia'''
* [[w:Affect_labeling|Affective labelling]] 
* [[Augmented Reality|Augmented reality]] 
* [[w:Brain-derived_neurotrophic_factor|Brain-derived neurotrophic factor]] 
* [[w:Broaden-and-build|Broaden-and-build theory]]
* [[w:Cognitive_processing_therapy|Cognitive processing therapy]] 
* [[w:Virtual_reality_sickness|Cyber sickness]]
* [[w:DSM-5|DSM-5]]
* [[w:Emotional_dysregulation|Emotional dysregulation]] 
* [[w:Extended_reality|Extended reality]]
* [[w:Malleability_of_intelligence|Malleability of intelligence]] 
* [[w:Memory_consolidation|Memory reconsolidation]]
* [[w:Mixed_reality_game|Mixed reality]] 
* [[w:Post-traumatic_stress_disorder|Post-traumatic stress disorder]]
* [[w:Prolonged_exposure_therapy|Prolonged exposure therapy]] 
* [[wikipedia:Virtual_reality_therapy|Virtual reality exposure therapy]]

== References ==
{{Hanging indent|Bailey, A. L., Kirsh, S., Rawlins, C., Persky, S., &amp; Clancy, C. (2024). Early scaling of immersive technology within the Veterans Health Administration. ''NEJM Catalyst Innovations in Care Delivery'', 5(4). https://doi.org/10.1056/cat.23.0356

Boska, R. L., Bishop, T. M., Capron, D. W., Paxton Willing, M. M., &amp; Ashrafioun, L. (2025). Difficulties with emotion regulation within PTSD clusters and moral injury subtypes. ''Military Psychology'', 37(2), 159-167. https://doi.org/10.1080/08995605.2024.2322904

de Haart, R., Daniels, J. K., Timmerman, M. E., Cath, D. C., &amp; Lommen, M. J. J. (2026). Augmenting virtual reality exposure for PTSD with physical activity: Study protocol of a randomised controlled trial. ''European'' ''Journal of Psychotraumatology'', 17(1), 2605803. https://doi.org/10.1080/20008066.2025.2605803

Efremov, A. (2025). Age-specific mental health profiles of combat veterans: Post-traumatic stress disorder and related disorders. ''Journal of Rational-Emotive &amp; Cognitive-Behavior Therapy'', 44(4), 1-15. https://doi.org/10.1007/s10942-025-00637-7

Elklit, A., &amp; Dahl, N. H. (2025). Emotion regulation difficulties, aggression, and PTSD symptoms in Danish treatment-seeking veterans. ''Scandinavian Journal of Military Studies'', 8(1), 308-326. https://doi.org/10.31374/sjms.264

Felemban, R. G., Alzahrani, R. R., Alrefaei, N. F., Alharbi, N. M., Alghamdi, A. S., &amp; Alqadi, S. (2026). Efficacy of virtual reality-based exposure therapy for post-traumatic stress disorder in military veterans: A systematic review and meta-analysis. ''Frontiers in Psychiatry'', 17, 1857109. https://doi.org/10.3389/fpsyt.2026.1857109

Fredrickson, B. L. (2001). The role of positive emotions in positive psychology: The broaden-and-build theory of positive emotions. ''American Psychologist'', 56(3), 218-226. https://doi.org/10.1037/0003-066X.56.3.218

Garrett, B., Taverner, T., Gromala, D., Tao, G., Cordingley, E., &amp; Sun, C. (2018). Virtual reality clinical research: Promises and challenges. ''JMIR Serious Games'', 6(4), e10839. https://doi.org/10.2196/10839

Kukharuk, O., Tkalich, K., Kamash, N., &amp; Georgiou, O. (2025). Effectiveness of immersive VR therapy in reducing stress-associated symptoms in Ukraine. ''European Journal of Psychotraumatology'', 16(1), 2488097. https://doi.org/10.1080/20008066.2025.2488097

Lewis, C., Roberts, N. P., Andrew, M., Starling, E., &amp; Bisson, J. I. (2020). Psychological therapies for post-traumatic stress disorder in adults: Systematic review and meta-analysis. ''European Journal of Psychotraumatology'', 11(1), 1729633. https://doi.org/10.1080/20008198.2020.1729633

Lopes, M. K. S., Perreault, L., de Jesus, B. Jr., Roberge, M. C., &amp; Falk, T. H. (2025). Subjective and objective evaluation of the benefits of multisensory virtual nature immersion for patients with post-traumatic stress disorder. In ''Proceedings of the 17th International Conference on Quality of Multimedia Experience'' (QoMEX) (pp.1-5). IEEE. https://doi.org/10.1109/QoMEX65720.2025.11219945

López-Ojeda, W., &amp; Hurley, R. A. (2022). Extended reality technologies: Expanding therapeutic approaches for PTSD. ''The Journal of Neuropsychiatry and Clinical Neurosciences'', 34(1), 1-5. https://doi.org/10.1176/appi.neuropsych.21100244

Macey, A.-L., Macey, J., &amp; Hamari, J. (2026). Emotion regulation in immersive virtual reality environments: A scoping review. ''Interacting with Computers'', 29, 1-20. https://doi.org/10.1093/iwc/iwag029

Niles, B., Lang, A., &amp; Olff, M. (2023). Complementary and integrative interventions for PTSD. ''European Journal of Psychotraumatology'', 14(2), 2247888. https://doi.org/10.1080/20008066.2023.2247888

Osman, A., Paczynski, M., &amp; Jha, A. P. (2016). Affective expectations influence neural responses to stressful images in soldiers. ''Military Psychology'', 29(1), 41-57. https://doi.org/10.1037/mil0000128

Selvakumaran, R. V. (2025). Developing virtual reality (VR) simulations with embedded user analytics for cognitive rehabilitation in PTSD veterans. In ''Proceedings of the 27th International Conference on Multimodal Interaction'' (pp. 740-744). ACM. https://doi.org/10.1145/3716553.3750826

Trejo, B. C., Richard, E. M., van Driel, M., &amp; McDonald, D. P. (2015). Cross-cultural competence: The role of emotion regulation ability and optimism. ''Military Psychology'', 27(5), 276-286. https://doi.org/10.1037/mil0000081

van Gelderen, M. J., Nijdam, M. J., &amp; Vermetten, E. (2018). An innovative framework for delivering psychotherapy to patients with treatment-resistant posttraumatic stress disorder: Rationale for interactive motion-assisted therapy. ''Frontiers in Psychiatry'', 9, 176. https://doi.org/10.3389/fpsyt.2018.00176

van Toorenburg, M. M., Sanches, S. A., Linders, B., Rozendaal, L., Voorendonk, E. M., Van Minnen, A., &amp; De Jongh, A. (2020). Do emotion regulation difficulties affect outcome of intensive trauma-focused treatment of patients with severe PTSD? ''European Journal of Psychotraumatology'', 11(1), 1724417. https://doi.org/10.1080/20008198.2020.1724417

Vermetten, E., Barcaro, S., Espejo, E., Bellini, P., Roy, M. J., &amp; Bremault-Phillips, S. (2025a). Linguistic analysis of patients’ labels during 3MDR psychotherapy. ''Psychiatry and Clinical Psychopharmacology'', 35(Suppl. 1), S29. https://doi.org/10.5152/pcp.2025.241024

Vermetten, E., Burback, L., Sevigny, P. R., Nijdam, M. J., Winkler, O., Espejo, E., Sessoms, P., Bright, K., Roy, M. J., &amp; Brémault-Phillips, S. (2025b). Brief manual for multi-modal motion-assisted memory desensitization and reconsolidation therapy for the treatment of post-traumatic stress disorder. ''Psychiatry and Clinical Psychopharmacology'', 35(Suppl. 1), S122. https://doi.org/10.5152/pcp.2025.241028

Westphal, M., Aldao, A., &amp; Jackson, C. (2017). Emotion dysregulation in comorbid posttraumatic stress disorder and substance use disorders: A narrative review. ''Military Psychology'', 29(3), 216-233. https://doi.org/10.1037/mil0000157

Wiederhold, B. K., &amp; Wiederhold, M. D. (2025). Virtual reality therapy combined with physiological monitoring provides effective treatment, with objective metrics, for post-traumatic stress disorder. ''Expert Review of Medical Devices'', 22(2), 117-119. https://doi.org/10.1080/17434440.2025.2454930
}}


== External links ==
* '''Web:''' [https://www.healthdirect.gov.au/post-traumatic-stress-disorder-ptsd Post traumatic stress disorder] (Australian Government)
* '''Web:''' [https://www.ptsd.va.gov/index.asp National Center for PTSD information home page]  (US Government)
* '''Web:''' [https://www.innovation.va.gov/hil/views/immersive/immersive-programs.html US Veterans Affairs Immersive Programs Innov]&lt;nowiki/&gt;[https://www.innovation.va.gov/hil/views/immersive/immersive-programs.html ation] (US Government)
* '''Web:''' [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Australian Royal Commission into Defence and Veteran Suicide - Final Repor]&lt;nowiki/&gt;[https://defenceveteransuicide.royalcommission.gov.au/publications/final-report t] (Australian Government)
* '''Web:''' [https://www.strategymrc.com/report/virtual-reality-therapy-for-ptsd-market Virtual reality therapy for PTSD market forecasts to 2032:] (Stratistics MR&lt;nowiki/&gt;C, a 2025 industry market report). 
* '''Podcast:''' [https://shows.acast.com/5c3353e484e2e79370e1d135/5c3353ef84e2e79370e1d13c PTSD Podcast] (Peace of Mind: Mental Health and Psychiatry, ACAST, 43 min)
* '''Video:''' [https://www.youtube.com/watch?v=bD43R_oa6qo 3MDR: Virtual reality treatment for veterans] (National Centre for Mental Health,Youtube, 2:46 min)
* '''Video:'''  [https://www.youtube.com/watch?v=jL2bKmniMTc VR exposure for combat PTSD] (PsyTech VR, Youtube, 2:20 min)  
* '''Final Report:''' [https://www.ncmh.info/wp-content/uploads/2020/05/Cardiff-3MDR-Study-Final-Report-with-cover-21.5.20.pdf 3MDR randomised control trial - Final-Report] (Cardiff University)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Psychotherapy]]
[[Category:Motivation and emotion/Book/Trauma]]
&lt;references /&gt;</text>
      <sha1>i0zsk9q2k5j5rakfz7y02hpb6l0wqi6</sha1>
    </revision>
    <revision>
      <id>2834786</id>
      <parentid>2834783</parentid>
      <timestamp>2026-09-28T03:57:20Z</timestamp>
      <contributor>
        <username>StretchBeyond</username>
        <id>3105744</id>
      </contributor>
      <minor />
      <comment>/* What does the research evidence show? */ Added Table 1 back into text.</comment>
      <origin>2834786</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35097" sha1="rpxc9r3vkqbqjw09rfo0vfeogb3eaf4" xml:space="preserve">{{title|Immersive therapy for PTSD treatment:&lt;br&gt;How does it work and what are the effects?}}
__TOC__
== Overview ==
{{RoundBoxTop|theme=2}}
'''Scenario: Immersive PTSD therapies'''

Andriy, a soldier, harnessed to a treadmill, walks towards an image he spent months avoiding, his clinical therapist  beside him. This is multi-modular motion assisted memory desensitisation and reconsolidation (3MDR), one of a new generation of immersive therapies for post-traumatic stress disorder. 
[[File:Andriy 3MDR enhanced.gif|center|600x600px]]

'''Figure 1.''' Andriy moves through his 3MDR treatment.

Learn about immersive therapy and Andriy’s* experience below (*Andriy: a fictional name). {{RoundBoxBottom}}
[[File:Post-traumatic_stress_disorder_world_map_-_DALY_-_WHO2004.svg|alt=|thumb|271x271px|'''Figure 2:''' The World Health Organisation (2024) estimated that [https://www.who.int/news-room/fact-sheets/detail/post-traumatic-stress-disorder 3.9% of the world's population had PTSD] at some stage.]]
[[w:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (PTSD) develops after severe or life-threatening trauma and carries a substantial personal and societal cost (Figure 2), with military personnel disproportionately represented (Boska et al., 2025). See costs and limitations below (Davis et al., 2022; Montgomery-Marks et al., 2025). 

PTSD's emotional impact is shaped by [[w:Emotional_dysregulation|emotional dysregulation]] - difficulty managing intense feelings such as guilt, fear or shame (Westphal et al., 2017). This commonly triggers [[Cognitive psychology|cognitive]] and behavioural avoidance that offers short-term relief but prevents traumatic memory from being adaptively processed, trapping individuals in cycles of avoidance and chronic hyperarousal (de Haart et al., 2026; van Gelderen et al., 2018).

Immersive interventions, including [[w:Virtual_reality_therapy|virtual reality exposure therapy]] (VRET) and 3MDR, aim to break this avoidance cycle by creating controlled environments in which trauma cues can be safely approached rather than avoided (van Gelderen et al., 2018; Wiederhold &amp; Wiederhold, 2025). This chapter explains the psychological theory behind these approaches, reviews the research evidence for their effects and considers their limitations.

{{RoundBoxTop|theme=2}}
'''Focus questions'''[[File:Crystal Clear app ktip.svg|left|20px|]]

* Why is emotional processing important in PTSD?
* How can immersive therapies influence the emotional processes underlying PTSD? 
* What does the research evidence show?
* What are the costs and limitations of immersive therapies? 
{{RoundBoxBottom}}

== Why is emotional processing important in PTSD? ==
[[File:PTSD.png|thumb|'''Figure 3.''' PTSD can deeply affect our emotions.]]PTSD is formally diagnosed according to [[w:DSM-5|DSM-5]] criteria and is characterised by disrupted executive and emotional processing systems, heightened threat perception, hypervigilance and persistent negative emotional states as depicted in Figure 3 (Kukharuk et al., 2025; López-Ojeda &amp; Hurley, 2022; Osman et al., 2016). Emotional processing theory provides the primary framework for understanding this disruption, supported by emotion regulation theory and inhibitory learning models introduced later in the chapter.

{{RoundBoxTop|theme=3}}[[Image:Crystal Clear app help index.svg|left|50px]]
;Predict the outcome

A soldier with PTSD encounters a trauma-related image expecting:

'''TRAUMA CUE → DANGER → DISTRESS → AVOID'''

However, during immersive treatment, the expected danger does not occur. What is the most likely consequence?&lt;quiz display=simple&gt;
{
|type=&quot;()&quot;}
- Fear increases permanently.

- Memory cannot change.

+ Prediction error creates opportunities for new learning.

- Emotional processing stops. 

}

&lt;/quiz&gt;
&lt;div style=&quot;text-align:right; color:red; font-weight:bold;&quot;&gt;
Click &quot;show&quot; below  to understand more⤵ &lt;/div&gt;
{{Hidden begin|title=Please consider the answer before opening this section}}The answer is '''C'''.
&lt;div style=&quot;text-align:left; color:black; font-weight:regular;&quot;&gt;
Mismatches between expected and actual danger create a
'''prediction error'''. This may contribute to fear extinction,
emotion regulation, and memory reconsolidation. '''Think about it:''' If approaching rather than avoiding trauma can create new learning, what role might emotion regulation, fear extinction, prediction error and memory reconsolidation play?
&lt;/div&gt;

 
{{Hidden end}}
&lt;div style=&quot;text-align:left; color:black; font-weight:regular;&quot;&gt;
&lt;/div&gt;{{RoundBoxBottom}}

=== Understanding PTSD and emotions ===
Emotional processing theory suggests recovery requires the trauma memory to be reactivated and updated with corrective information. In PTSD, this process is blocked and to manage distress, many individuals adopt cognitive and behavioural avoidance mechanisms (López-Ojeda &amp; Hurley, 2022). Avoidance offers short-term relief, but prevents the traumatic memory from being reactivated, so it cannot be updated. Trauma reminders such as flashbacks continue to trigger extreme distress, trapping the individual in a maladaptive, self-reinforcing avoidance cycle (Figure 4)  (Vermetten, Burback, et al., 2025b). 

At a neural level, Westphal et al. (2017) linked this to [[wikipedia:Transdiagnostic_process|transdiagnostic]] [[w:Emotional_dysregulation|emotion dysregulation]], in which the traumatic memory network remains isolated from the brain's [[w:Salience_network|salience]]  and central executive networks. Effective treatment requires reactivating this network so the memory can be integrated (Vermetten, Burback, et al., 2025b; Westphal et al., 2017). This is empirically supported: A 2025 study of Danish military veterans (''n''=142) found emotional regulation difficulties explained an additional 28% of the variance in PTSD symptoms; combined with [[w:Comorbidity|comorbid]] symptoms, these factors accounted for 52% of the variance in severity (''F''(13, 92) = 9.58, ''p'' &lt;0.001) (Elklit &amp; Dahl, 2025).[[File:Avoidance and Processing Cycles.png|500x500px|thumb|'''Figure 4'''. Avoidance cycles and the anticipated positive responses associated with immersive therapies. Based on: Lopez-Ojeda et al. (2022), Felemban et al. (2026), and Vermetten et. al. (2025b).|center]]

=== Why treatment can be difficult ===
Trauma-focused psychotherapies such as [[w:Prolonged_exposure_therapy|prolonged exposure]] (PE) and [[w:Cognitive_processing_therapy|cognitive processing therapy]] (CPT) ask patients to actively engage with distressing memories to generate fear extinction, precisely what avoidance prevents (van Toorenburg et al., 2020). Many patients cannot tolerate the emotional exposure these therapies require (Lopes et al., 2025; van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). Because the trigger is never confronted, the brain cannot experience a prediction error needed to learn the threat has passed, so symptoms persist (de Haart et al., 2026; López-Ojeda &amp; Hurley, 2022).

This is reflected in outcomes where an estimated 39.2% of patients fail to respond to standard trauma-focused therapy, and dropout rates range from 16-48% (de Haart et al., 2026; Vermetten, Burback, et al., 2025b). Among military veterans, younger patients show heightened severity when trauma is central to their identity, while others turn to poor diet or substance abuse, further eroding emotional regulation (Efremov, 2025; Niles et al., 2023).

[[w:Emotional_dysregulation|Emotional dysregulation]] was historically viewed as a fixed barrier requiring lengthy stabilisation before treatment could begin (van Toorenburg et al., 2020). More recent evidence suggests otherwise, although emotional regulation contributes to PTSD symptoms, it is also being increasingly viewed as a modifiable process with dynamic capacity, one able to improve as a natural consequence of successful memory processing (van Toorenburg et al., 2020). This reframes the clinical challenge as helping an individual safely approach and process the traumatic memory, not correcting a fixed deficit. [[File:UC PhD VR study.png|right|thumb|'''Figure 5.''' PhD work by Selvakumaran (University of Canberra) examining cultural adaptation using the US-''Bravemind'' system.]]Cultural context adds another layer of difficulty. Doctoral research at the [[University of Canberra]] is integrating exposure therapy with gamified, posture-adaptive cognitive rehabilitation to support physical and emotional recovery (Figure 5). Selvakumaran (2025), is examining how existing protocols such as used in the US-centric ''[https://medvr.ict.usc.edu/projects/bravemind.html Bravemind]'' system need cultural adaptation for Australian veterans and first responders, because their operational context and backgrounds differ from their US counterparts (Selvakumaran, 2025). These treatment gaps carry a substantial economic and personal cost, part of the motivation for developing alternatives such as immersive therapies.   

== How can immersive therapies influence emotional processes? ==
Immersive PTSD treatments mark a shift from passive, sedentary therapy towards active, embodied, interactive approaches (van Gelderen et al., 2018). Immersive therapy uses three psychological mechanisms and their interaction: multisensory presence, embodied cognition, and divergent thinking (López-Ojeda &amp; Hurley, 2022; van Gelderen et al., 2018).

Embodied cognition is how physical states of the body can directly modify states of the mind (van Gelderen et al., 2018). Immersion therapy goes beyond simple visual replication of a trauma memory; instead, capturing the participant’s visceral and cognitive focus by limiting distractions (Macey et al., 2026). 

=== What is immersive therapy? ===
[[w:Immersion_therapy|Immersive therapies]] use [[w:Extended_reality|extended reality]] (XR) platforms, encompassing [[virtual reality]] (VR), [[Augmented Reality|augmented reality]] (AR), and [[w:Mixed_reality_game|mixed reality]] (MR), to create customisable, controlled, and standardised therapeutic environments (López-Ojeda &amp; Hurley, 2022; Wiederhold &amp; Wiederhold, 2025). This ecosystem is depicted in Figure 6.
This chapter focuses on two applications: VRET where therapists reconstruct traumatic scenarios in safe, graded environments, and 3MDR, which extends this by having the patient move on a treadmill beside their therapist, rather than in a stationary, face-to-face session (de Haart et al., 2026; Felemban et al., 2026).[[File:XR and Human Senses.png|thumb|'''Figure 6.''' The Extended Reality environment. |center|750x750px]]
=== Presence and embodied cognition ===
Immersive therapy’s distinguishing feature is its ability to generate presence, or the psychological illusion of being ‘''there’''. The illusion is amplified by integrating synchronised audio, visual, [[w:Olfactory|olfactory]], [[wikipedia:Haptic_perception|haptic]], and movement stimuli (Lopes et al., 2025; López-Ojeda &amp; Hurley, 2022). This increases engagement with the trauma memory and supports emotional processing (van Gelderen et al., 2018).

In 3MDR, presence combines with cognition. The working principle is that physical states of the body can directly shape states of the mind (van Gelderen et al., 2018). Walking towards a virtual trauma reminder functions as a fear antagonistic action and rather than retreating in avoidance, the patient approaches, converting passive helplessness into active, empowered participation and disrupting the rigid and repetitive trauma narratives  common in PTSD (de Haart et al., 2026; Osman et al., 2016; van Gelderen et al., 2018). 

Notably, extinction learning normally requires moderate-to-high-intensity activity to stimulate [[w:Brain-derived_neurotrophic_factor|brain-derived neurotrophic factor]] (BDNF), but 3MDR's walking pace (&lt; 4 km/h) is too slow to generate meaningful BDNF secretion, making a major exercise induced BDNF mechanism unlikely, although physiological factors cannot be ruled out. The approach action and the cognitive restructuring it enables may be more important psychological drivers (de Haart et al., 2026).

=== Prediction error and inhibitory learning ===
Immersive therapy builds on the inhibitory learning model of exposure therapy, in which a new, safe association competes with and suppresses the original fear response. Walking towards a trauma cue and encountering safety instead of the expected catastrophe creates a prediction error between the anticipated, life-threatening event and actual reality. This may destabilise the traumatic memory, potentially creating conditions for memory reconsolidation to occur (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b; Wiederhold &amp; Wiederhold, 2025).

=== Memory reconsolidation ===
According to [[w:Memory_consolidation|memory reconsolidation]] theory, traumatic memories retrieved in a safe immersive context can become [[w:Malleability_of_intelligence|malleable]], allowing new, safe information to reconsolidate the memory in a non-threatening form (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). To prevent the patient from becoming overwhelmed, 3MDR uses dual-attention tasks such as tracking an oscillating ball (Figure 1), which tax working memory resources and reduce the vividness and emotional intensity of the memory (Vermetten, Burback, et al., 2025b).

Emerging linguistic research suggests this processing is reflected in patients' language. Across successive 3MDR sessions, [[w:Affect_labeling|affective labelling]] of feelings such as guilt shifted from past-tense to present-tense narration, consistent with a renewed ability to integrate traumatic memories into present-moment awareness (Vermetten, Barcaro, et al., 2025a).

== What does the research evidence show? ==
Addressing these barriers is important for any PTSD population, but especially critical for military populations with high treatment failure and dropout rates (van Gelderen et al., 2018). Immersive therapies as outlined in Table 1, can tailor patient-selected trauma cues to improve access to traumatic memory networks (Vermetten, Burback, et al., 2025b). 

=== Virtual reality exposure therapy ===
[[w:Virtual_reality_therapy|VRET]] reconstructs traumatic events in a structured context (Felemban et al., 2026; López-Ojeda &amp; Hurley, 2022). This may help bypass imagination challenges such as emotional numbing or amnesia that can prevent patients from engaging with traditional therapy (Macey et al., 2026). A meta-analysis of VRET for PTSD found substantial symptom reductions, averaging a 33.73-point decrease in the 0-80 point [[w:Clinician_Administered_PTSD_Scale|Clinician-Administered PTSD Scale]] and a 20.96-point decrease in the 0-80 point [https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PTSD Checklist PCL-5) scale]  (Felemban et al., 2026). Because changes of 10-20 points on these scales are considered clinically significant, this could mean the difference between severe functional impairment and mild or subclinical symptoms (Boska et al., 2025; de Haart et al., 2026; Felemban et al., 2026). Comparative effects against other active PTSD treatments remain modest, but VRET appears to be a more engaging alternative to conventional treatment (Felemban et al., 2026).

=== 3MDR ===
3MDR takes the same multisensory effect in VRET and adds an activating context. Rather than a sedentary, face-to-face session, the patient and the therapist face the virtual display together (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). It follows a three-phase protocol: pre-platform preparation, platform treadmill exposure, and post-platform re-consolidation (Vermetten, Burback, et al., 2025b). In a trial involving treatment-resistant PTSD; 3MDR showed large effect sizes from pre-treatment to six-months (''n'' = 134, ''d'' = 1.0) and high-acceptability, with 7-20% dropout rates (de Haart et al., 2026; van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). Substantially lower than the 16-48% typical trauma-focused therapy in military populations (Lewis et al., 2020).

Improvements are not limited to PTSD symptoms. A trial of 62 adults with severe PTSD, including childhood sexual trauma, found intensive trauma-focused treatment improved emotion-regulation abilities regardless of PTSD outcome (van Toorenburg et al., 2020). Some researchers link this improvement to positive psychology [[w:Broaden-and-build|broaden-and-build theory]]. Theorising that as patients regain a sense of safety and control, this may support a positive spiral of emotional flexibility that reinforces recovery (Fredrickson, 2001; Niles et al., 2023; Westphal et al., 2017).  

Critically, the exact causation remains unknown. Given the recent innovations in immersive therapies and small study samples, it remains unclear whether they work through exposure-based extinction, enhanced emotional engagement, physiological factors, embodiment or positive cognitive restructuring. These questions are relevant to both basic and applied research and have not yet been adequately addressed by current trial designs.  

'''Table 1'''

''Treatment Effects and Psychological Mechanisms''
{| class=&quot;wikitable mw-collapsible&quot;
| valign=&quot;top&quot; |'''Clinical Dimension'''
| valign=&quot;top&quot; |'''Traditional'''
| valign=&quot;top&quot; |'''Immersive'''
| valign=&quot;top&quot; |'''Psychological Mechanisms'''
|-
| valign=&quot;top&quot; |'''Context'''
| valign=&quot;top&quot; |'''''Sedentary'''''. Face-to-face, verbally describes trauma.
| valign=&quot;top&quot; |'''''Activating.''''' Dynamic, multi-sensory environment.
| valign=&quot;top&quot; |'''''Fear Antagonistic Action.'''''
'''''Approach behaviours.'''''

'''''Prediction Errors.'''''
|-
| valign=&quot;top&quot; |'''Trauma cue delivery'''
| valign=&quot;top&quot; |'''''Imaginary Retrieval'''''
Patient capacity
| valign=&quot;top&quot; |'''''Multisensory Immersion'''''
Highly tailored
| valign=&quot;top&quot; |'''''External Scaffolding.'''''
Bypasses internal barriers to activate memory networks.
|-
| valign=&quot;top&quot; |'''Processing'''
| valign=&quot;top&quot; |'''''Convergent processing.'''''
Repeated narration and fear habituation.
| valign=&quot;top&quot; |'''''Active Narrative Processing.'''''
Interactive, real-time affective labelling and dual attention tasks.
| valign=&quot;top&quot; |'''''Working Memory.'''''
Memory taxation reduces vividness and emotional intensity.
|-
| valign=&quot;top&quot; |'''Engagement''' 
| valign=&quot;top&quot; |'''''Attrition.'''''
Dropout rates 16-48%.
| valign=&quot;top&quot; |'''''Acceptability.'''''
Attractive. Dropout rates 7-20%.
| valign=&quot;top&quot; |'''''Sustained Motivation.'''''
Presence and safety in immersive environment.
|}
=== Applied example: 3MDR treatment in Ukraine ===
{{RoundBoxTop|theme=2}}
'''Scenario: Ukraine War: an applied 3MDR example'''

Andriy*, a Ukrainian Armed Forces volunteer, is struggling with hyper-vigilance, anxiety, and depression (Kukharuk et al., 2025). Traditional ‘talk’ therapy feels impossible; his mind remains in a constant state of combat readiness, even in quiet rooms.

In the pre-platform phase, Andriy identifies ‘hotspot’ memories in photographs. On the treadmill, harnessed and walking, he faces a panoramic screen as personalised warm-up music plays, selected to keep him connected to his traumatic memory network (Vermetten et al., 2025b). As his hotspot image fills the screen, his therapist asks three questions (Vermetten et al., 2025b):

1. What do you '''SEE''' ?
2. What does it '''TELL''' you?
3. What do you '''FEEL''' in your body NOW?

When Andriy identifies a surge of shame, the word '''GUILT''' is displayed as an affective label, and a dual-attention task begins. He tracks an oscillating, numbered ball while he stays with the emotion, taxing his working memory and reducing the intensity of the recalled trauma.
[[File:Andriy 3MDR enhanced.gif|center|600x600px]]

'''Figure 7.''' Andriy's 3MDR treatment.

By the end of the session, Andriy had walked toward what he used to avoid, creating a mismatch between his expectation of threat and his safety and turning a rigid, stuck memory into a manageable narrative.

*''Andriy, a fictional name given to one of 69 Ukrainian veterans, participating in a 2023 randomised controlled trial. Many demobilised after one year because of mental or neurological injuries'' (Kukharuk et al., 2025). 

{{RoundBoxBottom}}

== What are the costs and limitations of immersive therapies? ==
PTSD carries a substantial economic burden. An estimated US$232 billion in excess costs in the United States in 2018 and over £40 billion, in the United Kingdom in 2020-21, 92.4% of which were indirect rather than direct clinical costs (Davis et al., 2022; Montgomery-Marks et al., 2025). In Australia, the annual cost of PTSD per military veteran was estimated at $112,172 in 2025 (Magnusson &amp; Dey, 2025). In addition, individual healthcare costs surge by 142% in the year following a PTSD diagnosis, with comorbidities tripling this effect (Bothe et al., 2020). These figures span several countries and years and should be read as an indication of scale rather than as directly comparable totals.

Immersive therapy’s ability to move from small trials into mainstream PTSD treatment remains limited by methodological [[w:Homogeneity_and_heterogeneity|heterogeneity]], small sample sizes, and a lack of long-term data (Felemban et al., 2026). Many trials rely on military-heavy, treatment-resistant samples with few active comparison conditions, which limits the generalisation of findings to civilian or initial treatment populations. Practical issues such as [[w:Virtual_reality_sickness|cyber sickness]] or motion sickness can disrupt participation (Kukharuk et al., 2025).

Structural barriers include workforce training and equipment costs. Basic VR systems cost around US$3,500 per provider headset annually, with advanced simulation environments costing up to US$200,000 (Garrett et al., 2018). Despite this, adoption of immersive therapy is scaling. The [[w:United_States_Department_of_Veterans_Affairs|United States Veterans Affairs]] have expanded VR use from five medical centres in 2017 to over 154 centres and 2,300 staff, with applications now including over 40 documented clinical interventions such as chronic pain and suicide intervention (Bailey et al., 2024). Reflecting this potential growth trend, some market analysts estimate the global PTSD-focused VR therapy market was worth US$1.59 billion in 2025, forecasted to reach US$5.94 billion by 2032, driven largely by growing mental health awareness and the absence of standard clinical protocols (Stratistics MRC, 2025, an industry market report).

These limitations do not undermine the case for immersive therapy, but they show its evidence base and infrastructure are still maturing. Demonstrating rigorously why these therapies work, rather than assuming their novelty accounts for their effects is essential to avoid misallocating resources to unproven interventions and may support an emerging shift towards greater emphasis on post-traumatic growth and renewed optimism (Trejo et al., 2015; Vermetten, Burback, et al., 2025b; Wiederhold &amp; Wiederhold, 2025).

{{Robelbox|theme=2|title=Quiz}}
&lt;quiz display=simple&gt;
{Which of the following is an expected outcome of immersive therapy ?
|type=&quot;()&quot;}


- Patients are at risk because of an uncontrolled environment.
- The environments represent traditional face-to-face treatments.
- Failure and dropout rates are higher than traditional PTSD treatment.
+ Patients often broaden and build an upward spiral of emotion and optimism.
&lt;/quiz&gt;
{{Robelbox/close}}
== Conclusion ==
PTSD traps people in a self-reinforcing cycle of avoidance that blocks the adaptive processing necessary for traumatic memories to be resolved (see: ''[[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#Understanding PTSD and emotions|Understanding PTSD and emotions]]''). Standard trauma-focused therapies require patients to confront exactly what this cycle causes them to avoid, which contributes to high non-response and dropout rates (see ''[[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#Why treatment can be difficult|Why treatment can be difficult]]'').

Immersive therapies such as VRET and 3MDR address this by using presence, embodied cognition, prediction error and memory reconsolidation to help patients safely approach trauma cues rather than avoid them (see: ''[[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#What does the research evidence show?|What does the research evidence show]]''). The technology itself is only a delivery mechanism that enables the processes, the positive benefits appear to arise from the psychological mechanisms that immersive therapies facilitate, rather than the novelty of technology alone.

The evidence to date shows meaningful reductions in PTSD symptoms and comparatively low drop-out rates, alongside broader gains in emotional regulation. However, small samples, methodological variation, and a lack of long-term data mean the evidence base is still developing, and cost and infrastructure barriers remain significant (see: [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#What are the costs and limitations of immersive therapies?|costs and limitations]]'').{{RoundBoxTop|theme=11}}
[[File:Nuvola apps kuser.svg|Nuvola apps kuser|left|20px]] ''' Take-home message:''' 
Immersive therapies do not heal PTSD trauma through technological novelty or digital feedback. Instead, they create a safe, dynamic space that lets individuals confront trauma and reprocess memories into something they can live with. In doing so, people’s lives may once again shift toward value, optimism, and wellness.{{RoundBoxBottom}}

== See also ==
'''Wikiversity'''
* [[Motivation and emotion/Textbook/Emotion/Anxiety|Anxiety]] (Book chapter, 2010)
* [[/Non-invasive brain stimulation techniques/]] (Sub-page)
* [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)|Evidence-based PTSD assessment]]
* [[Motivation and emotion/Book/2024/Sense hacking|Sense hacking]] (Book chapter, 2024)

'''Wikipedia'''
* [[w:Affect_labeling|Affective labelling]] 
* [[Augmented Reality|Augmented reality]] 
* [[w:Brain-derived_neurotrophic_factor|Brain-derived neurotrophic factor]] 
* [[w:Broaden-and-build|Broaden-and-build theory]]
* [[w:Cognitive_processing_therapy|Cognitive processing therapy]] 
* [[w:Virtual_reality_sickness|Cyber sickness]]
* [[w:DSM-5|DSM-5]]
* [[w:Emotional_dysregulation|Emotional dysregulation]] 
* [[w:Extended_reality|Extended reality]]
* [[w:Malleability_of_intelligence|Malleability of intelligence]] 
* [[w:Memory_consolidation|Memory reconsolidation]]
* [[w:Mixed_reality_game|Mixed reality]] 
* [[w:Post-traumatic_stress_disorder|Post-traumatic stress disorder]]
* [[w:Prolonged_exposure_therapy|Prolonged exposure therapy]] 
* [[wikipedia:Virtual_reality_therapy|Virtual reality exposure therapy]]

== References ==
{{Hanging indent|Bailey, A. L., Kirsh, S., Rawlins, C., Persky, S., &amp; Clancy, C. (2024). Early scaling of immersive technology within the Veterans Health Administration. ''NEJM Catalyst Innovations in Care Delivery'', 5(4). https://doi.org/10.1056/cat.23.0356

Boska, R. L., Bishop, T. M., Capron, D. W., Paxton Willing, M. M., &amp; Ashrafioun, L. (2025). Difficulties with emotion regulation within PTSD clusters and moral injury subtypes. ''Military Psychology'', 37(2), 159-167. https://doi.org/10.1080/08995605.2024.2322904

de Haart, R., Daniels, J. K., Timmerman, M. E., Cath, D. C., &amp; Lommen, M. J. J. (2026). Augmenting virtual reality exposure for PTSD with physical activity: Study protocol of a randomised controlled trial. ''European'' ''Journal of Psychotraumatology'', 17(1), 2605803. https://doi.org/10.1080/20008066.2025.2605803

Efremov, A. (2025). Age-specific mental health profiles of combat veterans: Post-traumatic stress disorder and related disorders. ''Journal of Rational-Emotive &amp; Cognitive-Behavior Therapy'', 44(4), 1-15. https://doi.org/10.1007/s10942-025-00637-7

Elklit, A., &amp; Dahl, N. H. (2025). Emotion regulation difficulties, aggression, and PTSD symptoms in Danish treatment-seeking veterans. ''Scandinavian Journal of Military Studies'', 8(1), 308-326. https://doi.org/10.31374/sjms.264

Felemban, R. G., Alzahrani, R. R., Alrefaei, N. F., Alharbi, N. M., Alghamdi, A. S., &amp; Alqadi, S. (2026). Efficacy of virtual reality-based exposure therapy for post-traumatic stress disorder in military veterans: A systematic review and meta-analysis. ''Frontiers in Psychiatry'', 17, 1857109. https://doi.org/10.3389/fpsyt.2026.1857109

Fredrickson, B. L. (2001). The role of positive emotions in positive psychology: The broaden-and-build theory of positive emotions. ''American Psychologist'', 56(3), 218-226. https://doi.org/10.1037/0003-066X.56.3.218

Garrett, B., Taverner, T., Gromala, D., Tao, G., Cordingley, E., &amp; Sun, C. (2018). Virtual reality clinical research: Promises and challenges. ''JMIR Serious Games'', 6(4), e10839. https://doi.org/10.2196/10839

Kukharuk, O., Tkalich, K., Kamash, N., &amp; Georgiou, O. (2025). Effectiveness of immersive VR therapy in reducing stress-associated symptoms in Ukraine. ''European Journal of Psychotraumatology'', 16(1), 2488097. https://doi.org/10.1080/20008066.2025.2488097

Lewis, C., Roberts, N. P., Andrew, M., Starling, E., &amp; Bisson, J. I. (2020). Psychological therapies for post-traumatic stress disorder in adults: Systematic review and meta-analysis. ''European Journal of Psychotraumatology'', 11(1), 1729633. https://doi.org/10.1080/20008198.2020.1729633

Lopes, M. K. S., Perreault, L., de Jesus, B. Jr., Roberge, M. C., &amp; Falk, T. H. (2025). Subjective and objective evaluation of the benefits of multisensory virtual nature immersion for patients with post-traumatic stress disorder. In ''Proceedings of the 17th International Conference on Quality of Multimedia Experience'' (QoMEX) (pp.1-5). IEEE. https://doi.org/10.1109/QoMEX65720.2025.11219945

López-Ojeda, W., &amp; Hurley, R. A. (2022). Extended reality technologies: Expanding therapeutic approaches for PTSD. ''The Journal of Neuropsychiatry and Clinical Neurosciences'', 34(1), 1-5. https://doi.org/10.1176/appi.neuropsych.21100244

Macey, A.-L., Macey, J., &amp; Hamari, J. (2026). Emotion regulation in immersive virtual reality environments: A scoping review. ''Interacting with Computers'', 29, 1-20. https://doi.org/10.1093/iwc/iwag029

Niles, B., Lang, A., &amp; Olff, M. (2023). Complementary and integrative interventions for PTSD. ''European Journal of Psychotraumatology'', 14(2), 2247888. https://doi.org/10.1080/20008066.2023.2247888

Osman, A., Paczynski, M., &amp; Jha, A. P. (2016). Affective expectations influence neural responses to stressful images in soldiers. ''Military Psychology'', 29(1), 41-57. https://doi.org/10.1037/mil0000128

Selvakumaran, R. V. (2025). Developing virtual reality (VR) simulations with embedded user analytics for cognitive rehabilitation in PTSD veterans. In ''Proceedings of the 27th International Conference on Multimodal Interaction'' (pp. 740-744). ACM. https://doi.org/10.1145/3716553.3750826

Trejo, B. C., Richard, E. M., van Driel, M., &amp; McDonald, D. P. (2015). Cross-cultural competence: The role of emotion regulation ability and optimism. ''Military Psychology'', 27(5), 276-286. https://doi.org/10.1037/mil0000081

van Gelderen, M. J., Nijdam, M. J., &amp; Vermetten, E. (2018). An innovative framework for delivering psychotherapy to patients with treatment-resistant posttraumatic stress disorder: Rationale for interactive motion-assisted therapy. ''Frontiers in Psychiatry'', 9, 176. https://doi.org/10.3389/fpsyt.2018.00176

van Toorenburg, M. M., Sanches, S. A., Linders, B., Rozendaal, L., Voorendonk, E. M., Van Minnen, A., &amp; De Jongh, A. (2020). Do emotion regulation difficulties affect outcome of intensive trauma-focused treatment of patients with severe PTSD? ''European Journal of Psychotraumatology'', 11(1), 1724417. https://doi.org/10.1080/20008198.2020.1724417

Vermetten, E., Barcaro, S., Espejo, E., Bellini, P., Roy, M. J., &amp; Bremault-Phillips, S. (2025a). Linguistic analysis of patients’ labels during 3MDR psychotherapy. ''Psychiatry and Clinical Psychopharmacology'', 35(Suppl. 1), S29. https://doi.org/10.5152/pcp.2025.241024

Vermetten, E., Burback, L., Sevigny, P. R., Nijdam, M. J., Winkler, O., Espejo, E., Sessoms, P., Bright, K., Roy, M. J., &amp; Brémault-Phillips, S. (2025b). Brief manual for multi-modal motion-assisted memory desensitization and reconsolidation therapy for the treatment of post-traumatic stress disorder. ''Psychiatry and Clinical Psychopharmacology'', 35(Suppl. 1), S122. https://doi.org/10.5152/pcp.2025.241028

Westphal, M., Aldao, A., &amp; Jackson, C. (2017). Emotion dysregulation in comorbid posttraumatic stress disorder and substance use disorders: A narrative review. ''Military Psychology'', 29(3), 216-233. https://doi.org/10.1037/mil0000157

Wiederhold, B. K., &amp; Wiederhold, M. D. (2025). Virtual reality therapy combined with physiological monitoring provides effective treatment, with objective metrics, for post-traumatic stress disorder. ''Expert Review of Medical Devices'', 22(2), 117-119. https://doi.org/10.1080/17434440.2025.2454930
}}


== External links ==
* '''Web:''' [https://www.healthdirect.gov.au/post-traumatic-stress-disorder-ptsd Post traumatic stress disorder] (Australian Government)
* '''Web:''' [https://www.ptsd.va.gov/index.asp National Center for PTSD information home page]  (US Government)
* '''Web:''' [https://www.innovation.va.gov/hil/views/immersive/immersive-programs.html US Veterans Affairs Immersive Programs Innov]&lt;nowiki/&gt;[https://www.innovation.va.gov/hil/views/immersive/immersive-programs.html ation] (US Government)
* '''Web:''' [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Australian Royal Commission into Defence and Veteran Suicide - Final Repor]&lt;nowiki/&gt;[https://defenceveteransuicide.royalcommission.gov.au/publications/final-report t] (Australian Government)
* '''Web:''' [https://www.strategymrc.com/report/virtual-reality-therapy-for-ptsd-market Virtual reality therapy for PTSD market forecasts to 2032:] (Stratistics MR&lt;nowiki/&gt;C, a 2025 industry market report). 
* '''Podcast:''' [https://shows.acast.com/5c3353e484e2e79370e1d135/5c3353ef84e2e79370e1d13c PTSD Podcast] (Peace of Mind: Mental Health and Psychiatry, ACAST, 43 min)
* '''Video:''' [https://www.youtube.com/watch?v=bD43R_oa6qo 3MDR: Virtual reality treatment for veterans] (National Centre for Mental Health,Youtube, 2:46 min)
* '''Video:'''  [https://www.youtube.com/watch?v=jL2bKmniMTc VR exposure for combat PTSD] (PsyTech VR, Youtube, 2:20 min)  
* '''Final Report:''' [https://www.ncmh.info/wp-content/uploads/2020/05/Cardiff-3MDR-Study-Final-Report-with-cover-21.5.20.pdf 3MDR randomised control trial - Final-Report] (Cardiff University)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Psychotherapy]]
[[Category:Motivation and emotion/Book/Trauma]]
&lt;references /&gt;</text>
      <sha1>rpxc9r3vkqbqjw09rfo0vfeogb3eaf4</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Machiavellian motivation</title>
    <ns>0</ns>
    <id>331037</id>
    <revision>
      <id>2834835</id>
      <parentid>2834110</parentid>
      <timestamp>2026-09-28T11:00:16Z</timestamp>
      <contributor>
        <username>Mim0502</username>
        <id>3106283</id>
      </contributor>
      <comment>started writing</comment>
      <origin>2834835</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="37891" sha1="ttawd6dbrcno9xuf2o4zjx9kr0zd8pb" xml:space="preserve">{{title|Machiavellian motivation:&lt;br&gt;What is the motivational role of Machiavellianism?}}
__TOC__
==Overview==

{{RoundBoxTop|theme=5}}
'''Imagine this...''' 

[[File:The death of Niccolò Machiavelli (1848), by Cesare Dell'Acqua.jpg|thumb|'''Figure 1'''. The death of Niccolo Machiavelli, the relevant historical figure famous for analysing quiet forms of power]]

In a collaborative project, one member, Darren, quietly positions himself as the 'team organiser'. He chooses the simple, high-visibility tasks for himself and assigns the more demanding, time-heavy work to quieter members. When speaking to the supervisor, he presents the team's progress in a way that highlights his coordination and softens the significance of everyone else's contributions. By the end, he is perceived as the natural leader. 

You feel a mix of frustration and discouragement, not because anything openly unfairly happened, but because the imbalance was created through subtle decisions that were difficult to challenge. It leaves you with a sense of being strategically sidelined, as if your effort mattered but was never meant to be recognised. 

Later, while reflecting on the experience, you recall one of your colleagues briefly mention the anniversary of a historical philosopher known for analysing quiet forms of influence and power. The reference wasn't connected to your project, yet it lingers. It makes you wonder whether Darren's behaviour was intentional or simply a familiar pattern he's learned to rely on. 
{{ic|The target audience is a global audience, so consider making the example more applicable in that context.}}

{{RoundBoxBottom}}


''author note: this overview is terribly cluttered at the moment, just jotting down notes/info I want to mention at the start then re-structuring it when I find it flows well from one point to the next''
* '''''Fig. 1:''' the death of Niccolo Machiavelli coincided with a profound moment of crisis in Italy, and immediately preceded the widespread dissemination of his work; he died having never seen his most famous work (The Prince), his realism, institutional thinking, and trust in the people further reveal a philosophy oriented toward popular participation, civic liberty and resistance to elite domination (Wills, 2021) {{ic|probably better off linking out to this detail in a Wikipedia article}}''
* ''It is important to understand dark-side traits in organisational and academic performance -''
* ''The concept of Machiavellian Motivation {{ic|use lower case for motivation}}, the core definition, machiavellianism as a personality trait - defined as using manipulation against other people's interests to achieve one's own personal goal (Aldousari &amp; Ickes, 2021)''
* ''the central argument, what does the chapter say/prove about machiavellianism {{ic|use upper case for Machiavellianism}} : Sprangler &amp; Tikhomirov (2025) argue that Machiavellian leaders embody &quot;personal power&quot;; self-focused, dominance-oriented motive that drives strategic manipulation and short-term effectiveness, and in contrast, &quot;socialised power&quot; leaders channel influence toward collective benefit, producing ethical sustainable organisational outcomes''
* ''drawing on the Dark Triad framework, self-determination theory, human motivation theory''
* ''motivational mechanisms, contextual application - refer back to scenario????, evidence, chapter structure''
''Social Cognitive Theory'' 

Why do some individuals pursue influence through manipulation rather than cooperation? This question lies at the centre of Machiavellianism, a personality trait characterised by strategic manipulation and the exploitation of others for personal gain (Aldousari &amp; Ickes, 2021). As a core component of the Dark Triad, Machiavellianism has attracted considerable attention within organisational psychology because of its implications for leadership, performance, and workplace relationships.  

This chapter examines the concept of Machiavellian motivation, referring to the motives and psychological processes that drive manipulative and self-serving behaviour. Although Machiavellian individuals often appear effective in competitive environments, this chapter argues that such effectiveness is largely short-lived. Specifically, it contends that Machiavellianism reflects a self-focused power orientation that may facilitate immediate gains but ultimately undermines trust, cooperation, and sustainable organisational outcomes. In doing so, the chapter challenges the notion that strategic manipulation is an effective long-term pathway to influence and success. This position aligns with Sprangler and Tikhomirov's (2025) distinction between personal power and socialised power, whereby Machiavellian leaders pursue influence for individual advantage, whereas socialised power leaders channel influence towards collective benefit, producing more ethical and sustainable organisational outcomes.  

''Drawing on the Dark Triad framework, Social Cognitive Theory, Self-Determination Theory, and broader theories of human motivation, the chapter explores the motivational foundations of Machiavellian behaviour and the mechanisms through which it influences organisational outcomes. It first defines Machiavellianism and situates it within the wider personality literature, before examining its motivational underpinnings, reviewing empirical evidence, and discussing implications for organisations and leaders.''   

Recommended length: 180 to 330 words.
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]

{{RoundBoxTop|theme=5}}
'''Focus questions'''

* How has Machiavellianism developed conceptually as a motivational construct?
* What are the motivational characteristics of Machiavellianism as a trait?
* What core motives drive Machiavellian individuals?
* How do Machiavellian motives translate into behavioural strategies?
{{RoundBoxBottom}}

== Origins and conceptual development of machiavellianism ==
'''FQ: How has Machiavellianism developed conceptually as a motivational construct?'''

''author note: I am aware some of these sources are not peer-reviewed sources, these are just preliminary.''

* &quot;Men are driven by necessity, and he who seeks to deceive will always find someone who will allow himself to be deceived&quot; (Machiavelli, 1532/2017) {{ic|APA style - direct quotes need page numbers.}}
* &quot;Men are driven by two principal impulses, either by love or by fear&quot; (Machiavelli, 1532/2017) - Machiavelli believed a ruler should ideally be both, but since human beings are unreliable, love is a weak foundation for authority.

=== Philosophical Roots -&gt; Historical and theoretical foundations ===

* Philosophies for Life. (2024): Niccolo Machiavelli - referred to as the &quot;father of modern political philosophy&quot;, wrote the novel 'The Prince'
** separated politics from ethics
** he believed people are naturally self-interested and unreliable
** 'virtu' = skill, cleverness (courage, wisdom, decisiveness) / 'fortuna' = luck, chance
** &quot;perception is everything&quot;
* Machiavellianism became a negative concept because readers (public &amp; religious reactions) misinterpreted Niccolo Machiavelli's 16th century political manual, The Prince, as an endorsement of deceit rather than a realistic portrayal of how power worked (MR. BRAIN, 2025) - Machiavelli's treatment of mortality is one of the most controversial aspects of his thought, 'the ends justify the means'
* He rejected the view of state existing to cultivate virtue and the good life, insisting that politics had to be considered/understood on its own terms guided by history and human psychology rather than religious doctrine or idealised notions of justice - he is often credited as an early voice of [https://www.britannica.com/topic/secularism Secularism] (Britannica) in Western political thought (Kumar, 2025)

* Core pillars on his political thought = Power, manipulation and strategic pragmatism (Kumar, 2025)
* &quot;This is to be asserted in general of men, that they are ungrateful, fickle, false, cowardly, covetous&quot; (Machiavelli, 1532/2017)
* &quot;The lion cannot defend itself against snares and the fox cannot defend itself against wolves&quot; (Machiavelli, 1532/2017)
* Connell (2022) blended intellectual history with biographical reconstruction to show how Machiavelli's end-of-life circumstances refract his lifelong preoccupation with power, fortune, and human nature

=== Evolution into a motivational trait ===

==== Transition to Modern Psychology -&gt; Integration into modern personality models ====
* Machiavellianism as a Spectrum Trait - Persson &amp; Kajonius (2017) provide definitive evidence that Machiavellianism is a spectrum trait; it varies continuously across individuals, with graded differences in cynicism, manipulation, and emotional detachment. It is dimensional and measurable, not categorical.
* Integration into the [[wikipedia:Dark_triad|dark triad]] framework
** Paulhus &amp; Williams (2002) establish the Dark Triad as a unified but differentiated framework of socially aversive personality; Machiavellianism, [[Narcissistic personality disorder|Narcissism personality disorder]] (Wikiversity), and [[wikipedia:Psychopathy|Psychopathy]] (Wikipedia) share a manipulative, callous interpersonal cure but diverge sharply in impulsivity, self-presentation, and strategic orientation
** Rauthmann (2012) identified stable behavioural signatures of Machiavellianism (manipulation, strategic planning, etc.) and provided structural evidence that Machiavellianism is a coherent trait, not a behavioural tendency

==== Empirical Foundations -&gt; Shift toward a motivational construct ====
* Christie &amp; Geis (1970) reached their conclusions by observing real manipulative behaviour, identifying the attitudes that predicted it, and using factor analysis to build a coherent trait scale.  
** They developed the [https://openpsychometrics.org/tests/MACH-IV/ Mach-IV Scale] (MACH-IV: Machiavellianism Test), foundational self-report instrument for measuring Machiavellianism in non-clinical population, operationalising Machiavellianism as a strategic and cynical interpersonal orientation, capturing both beliefs about human nature and preferred interpersonal tactics
* Establishing Machiavellianism as a Measurable Trait 
** Dahling et al. (2009) demonstrated that Machiavellianism is a measurable, multidimensional trait in workplace setting through their Workplace Machiavellianism Scale; which provides strong psychometric evidence that Machiavellianism is a stable dispositional construct, not merely a situational tactic

== Trait-based motivational tendencies of machiavellianism ==
'''FQ: What are the motivational characteristics of Machiavellianism as a trait?'''

''author note: this section, at most, is only at a fundamental level and I am still planning on breaking down into further subsections. the information/cites provided are only preliminary.''

Wilson et al. (1997) stated that Machiavellianism became a term of interest in evolutionary psychology

* it has been argued that manipulative behaviour is not a single trait but rather a complex set of traits that cannot be captured by a single scale 
* the article argues that Machiavellianism is an adaptive, context-sensitive strategy shaped by evolutionary pressures and expressed through psychological mechanisms that favour manipulation; its effectiveness may depend on environmental context and population dynamics
* Evolutionary theory explains why the trait exists 
Dark Triad Framework

* Veselka et al. (2012) show that the Dark Triad is best understood through the HEXACO model (see Fig. 2), where low Honesty-Humility forms the common foundation and additional trait combinations differentiate machiavellianism, psychopathy, and narcissism

=== Cognitive architecture of machiavellian motivation ===
[[File:HEXACO 1.png|thumb|'''Fig. 2''' Ashton &amp; Lee's HEXACO Model]]

==== Core Personality Characteristics (Nature of Personality) ====
* Machiavellianism strongly reflects low H-H in Ashton &amp; Lee's (2007) HEXACO model; characterised by exploitation, entitlement, and manipulation
* Christie &amp; Geis (1970) stated machiavellians demonstrated strategic, calculating pragmatism, or the preference for instrumental, goal-focused reasoning over moral or relational considerations
* High cognitive empathy and low affective empathy, or the ability to understand others' perspectives without emotional resonance (Wai &amp; Tiliopoulos, 2012)

==== Affective and Cognitive Style (Feeling &amp; Thoughts) ====
* Blunted moral emotions facilitate manipulation; low guilt, remorse and affective responsiveness (Ali &amp; Chamorro-Premuzic, 2010)
* Machiavellians relies on controlled processing rather than intuitive empathy; analytical social cognition (Jones &amp; Paulhus, 2011)
* Grieve &amp; Mahar (2010) found they suppressed or modulated emotional displays to maintain influence; emotion regulation for strategi ends

=== Instrumental and controlled goal orientation ===

==== Interpersonal Orientation (In relation to others) ====
* Dahling et al. (2008) discussed machiavellians to be skilled in navigating organisational politics and coalition building 
* Exploitative, transactional relational approach; wherein they only engage in relationships when instrumental returns are high (Kiazad et al., 2010)
* They use charm, deception, and impression management to achieve goals revealing a manipulative interpersonal style (Christie &amp; Geis, 1970; Jones &amp; Paulhus, 2009)
* White et al. (2024) found that machiavellianism is a meaningful predictor of intimate partner violence, particularly psychological and coercive forms
* Blötner &amp; Bergold (2023) show that cyberbullying is particularly appealing to Machiavellians because it allows them to harm other strategically while avoiding detection

==== Self-Determination Theory ====
Three basic psychological needs: Autonomy, relatedness, competence (Deci &amp; Ryan, 2000).

* McHoskey (1999) shows that Machiavellianism reflects a motivation profile dominated by extrinsic goals, low social interest, and controlled regulation

===== Extrinsic Orientation (Relatedness) =====
* Instrumental Motivation; Blötner &amp; Bergold (2022b) demonstrated that Machiavellianism contains both approach-oriented manipulation and avoidance-oriented distrust. These facets jointly determine whether Machiavellians choose to deceive, defend, or strategically oscillate between both behaviours
* External Self-concept motivation
* Moss &amp; Barbuto Jr. (2004) demonstrated that Machiavellians seek control and status using coercive, indirect and strategic tactics to influence others rather than rational or inspirational methods

===== Autonomy and Low Affiliation (and Competence) =====
* Rauthmann (2012) found that Machiavellians prefer autonomy, independence, and emotional distance
** showing low affiliative motivation and avoiding relational dependence

*Wilson et al. (1996) demonstrated that Machiavellians show low need for closeness and exhibits high competence-related self-efficacy
*Similarly, McHoskey (1995) reported low affiliation needs and high agentic motives such as power, control, and competence

== Core motives of machiavellian individuals ==
'''FQ: What core motives drive Machiavellian individuals?'''

David McClelland's (1965) Human Motivation Theory and Deci and Ryan's (2000) Self-Determination Theory together show that Machiavellians act the way they do because: 

* they are power-motivated (Human Motivation Theory) and extrinsically driven (SDT)
* they pursue goals that increase control, status, and advantage, not intrinsic satisfaction
* they avoid emotional closeness because affiliation and relatedness undermine strategic autonomy
* they invest in competence and achievement only when it enhances influence 
* their motivational system is instrumental and contex-sensitive, not prosocial or self-expressive

=== Approach motives: power, control, and strategic gain ===

==== Human Motivation Theory ====
McClelland's (1965) Human Motivation Theory: Achievement, Affiliation, Power

===== Power and Control – power motivation? - Power (nPow) =====
* Machiavellianism is positively associated with self-efficacy and job performance when individuals believe they can control outcomes, suggesting that feeling capable amplifies their motivation to pursue self-interested goals (Hafeez et al., 2024)
* Closely linked to [[Motivation and emotion/Book/2013/Power motivation|Power motivation]] (Power motivation, 2013) - people who are high machs view dominance and control as core personal objectives (Christie &amp; Geis, 1970)
* Szabo et al. (2023) identify power motivation as a central driver of Machiavellian work behaviour, part of their broader pattern of controlled, extrinsic, strategic motivation, which distinguishes Machiavellianism from the other Dark Triad traits.
* Aldousari &amp; Ickes (2021) found a positive correlation between machiavellianism and an external locus of control, which reflects a worldview where powerful others and situational forces dictate outcomes. This fuels the belief system that Machiavellians' relies on manipulation and strategic behaviour as compensatory mechanisms for perceive lack of personal control

===== Reward Sensitivity - Achievement (nAch) =====
* Buravlova (2026) demonstrates that Machiavellianism is neurally characterised by string reward sensitivity and high execute control, enabling calculated manipulation and strategic social behaviours
* Mercadante et al. (2026) show that managers sometimes evaluate dark-trait subordinates positively because these individuals facilitate agentic, performance-oriented goals; machiavellianism being especially rewarded due to its reward-sensitive, achievement-oriented profile
* Birkás et al. (2016) demonstrate that Machiavellianism is characterised by high reward sensitivity, low punishment sensitivity, and strategic risk-taking, aligning closely with achievement motivation (nAch) and explains Machiavellians' opportunistic, reward-driven behaviour

=== Avoidance motives and manipulation self-efficacy ===
Social cognitive theory self-efficacy

== Behavioural expression of machiavellian motivation ==
'''FQ: How do Machiavellian motives translate into behavioural strategies?'''

''author note: this section, at most, is only at a fundamental level and I am still planning on breaking down into further subsections. the information/cites provided are only preliminary.''

* Personality psychology through organisational behaviours to decision-making research underpins how machiavellian tendencies scale upward.
* Jahangir et al. (2024) show that Machiavellianism is supported by a coherent psychological and neural profile&quot; emotionally detached, strategically cognitive, clinically antagonistic, and nerually biased toward executive control over emotional processing

=== Strategic social influence - interpersonal behaviours ===

===== Strategic Calculation =====
* Long-term Planning; This multidimensional evidence from Jahangir et al. (2024) reinforces the view of Machiavellianism as a cold, calculated social strategt rather than an impulsive or purely antisocial trait
* Strategic Opacity - machiavellianism shapes decision-making not only through individual traits but through team dynamics and organisational structures as well, it is a systemic force that influences how information flows, how power is distributed and how strategic decisions are made within organisations (Matthews et al., 2022)
* Calculated Manipulation; Gao et al. (2025) show that antagonistic Machiavellians achieve greater promotability not because their antagonism is inherently valued, but because certain organisational contexts reward political behaviour and misinterpret strategic self-presentation as leadership potential

===== Instrumental Sociability =====
* Social Acuity; Jones (2016) shows that machiavellians misbehave in ways that are planned, covert and opportunistic. Their antisocial behaviour is a strategic tool for achieving personal gain, not an impulsive or emotional reaction
* Transactional Relationships 
* Utilitarian Focus; Genau et al. (2021) found that machiavellian leaders are effective only when they possess high political skill, acting as 'social lubricant' allowing machiavellian tendencies to be expressed in socially acceptable and effective ways. Without it, machiavellianism leads to dysfunctional and distrust-based leadership.

=== Context-shaped opportunistic behaviour - decision-making and situational behaviours ===

===== Deception and Leverage =====
* Geis &amp; Moon (1981) show that Machiavellian individuals deceive because it is instrumentally useful, emotionally easy, and cognitively manageable. Their deception is deliberate and aimed at securing advantage rather that driven by impulse or malice.
* Furthermore, Blötner &amp; Bergold's (2022a) central finding is that Machiavellian individuals are skilled &quot;bullshit (persuasive statements made without regard for truth) producers&quot;, but not reliably more resistant to bullshit themselves

* Emotional Detachment; Gao &amp; Fang (2025) found that Machiavellianism harms adolescent wellbeing not through direct emotional distress, but through its interpersonal consequences: reduced prosocial behaviour, increased loneliness, and heightened suicidal risk
* Delayed Gratification; Bratu et al. (2025) show that machiavellianism undermines academic performance because it promotes extrinsic motivation strategic lying, and low genuine engagement with learning. Machiavellian students pursue grades instrumentally rather than through mastery, which makes deception a functional (though academically harmful) strategy

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Machiavellianism is a measurable, multidimensional spectrum trait supported by decades of psychometric work, from the Mach-IV to modern taxometric analyses confirming its continuous latent structure
* Across behavioural, motivational, and neural evidence, Machiavellianism consistently aligns with high autonomy, low affiliation, and instrumental competence. 
** This forms a cold, agentic, interpersonal strategy oriented toward control and reward acquisition
* Empirical studies show that Machiavellians' reward sensitivity, strategic cognition, and emotional detachment hointly produce stable patterns of manipulation and opportunism, reinforcing the trait;s coherence across contexts.

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses

==References==
{{Hanging indent|1=

Aldousari, S. S., &amp; Ickes, W. (2021). How is Machiavellianism related to locus of control?: A meta-analytic review. Personality and Individual Differences, 174, 110677. https://doi.org/10.1016/j.paid.2021.110677

Aldousari, S. S., &amp; Ickes, W. (2021). How is Machiavellianism related to locus of control?: A meta-analytic review. Personality and Individual Differences, 174, 110677. https://doi.org/10.1016/j.paid.2021.110677

Ali, F., &amp; Chamorro-Premuzic, T. (2010). Investigating Theory of Mind deficits in nonclinical psychopathy and Machiavellianism. Personality and Individual Differences, 49(3), 169–174. https://doi.org/10.1016/j.paid.2010.03.027

Ashton, M. C., &amp; Lee, K. (2007). Empirical, Theoretical, and Practical Advantages of the HEXACO Model of Personality Structure. Personality and Social Psychology Review, 11(2), 150–166. https://doi.org/10.1177/1088868306294907

Beller, J., &amp; Bosse, S. (2017). Machiavellianism has a dimensional latent structure: Results from taxometric analyses. Personality and Individual Differences, 113, 57–62. https://doi.org/10.1016/j.paid.2017.03.014

Birkás, B., Csathó, Á., Gács, B., &amp; Bereczkei, T. (2015). Nothing ventured nothing gained: Strong associations between reward sensitivity and two measures of Machiavellianism. Personality and Individual Differences, 74, 112–115. https://doi.org/10.1016/j.paid.2014.09.046

Blötner, C., &amp; Bergold, S. (2022a). It is double pleasure to deceive the deceiver: Machiavellianism is associated with producing but not necessarily with falling for bullshit. British Journal of Social Psychology, 62(1), 467–485. https://doi.org/10.1111/bjso.12559

Blötner, C., &amp; Bergold, S. (2022b). To be fooled or not to be fooled: Approach and avoidance facets of Machiavellianism. Psychological Assessment, 34(2), 147–158. https://doi.org/10.1037/pas0001069

Blötner, C., &amp; Bergold, S. (2023). The Machiavellian bully revisited: A closer look at differences and processes of Machiavellian bullying and cyberbullying perpetration. Aggressive Behavior, 49(6), 568–579. https://doi.org/10.1002/ab.22095

Bratu, M. L., Rosca, L. I., &amp; Rosca, N. A. (2025). Machiavellianism, Lying, and Motivation as Predictors of Academic Performance in Romanian Engineering Students. Education Sciences, 15(8), 1028. https://doi.org/10.3390/educsci15081028

Buravlova, A. (2026). The manipulation map: How the Dark Triad shapes the brain. Psychiatry Research: Neuroimaging, 360, 112227. https://doi.org/10.1016/j.pscychresns.2026.112227

Christie, R., &amp; Geis, F. L. (1970). Studies in Machiavellianism. Academic Press.

Connell, W. (2022). The Secretary’s Last Rites: Machiavelli On His Deathbed. Machiavelliana, 1, Pp. 61-80 . https://doi.org/10.19272/202216101003

Dahling, J. J., Whitaker, B. G., &amp; Levy, P. E. (2008). The Development and Validation of a New Machiavellianism Scale. Journal of Management, 35(2), 219–257. https://doi.org/10.1177/0149206308318618

Gao, K., &amp; Fang, F. (2025). Longitudinal impact of machiavellianism on adolescent prosocial behavior and suicidal risk: the role of hope and loneliness. BMC psychology, 13(1), 493. https://doi.org/10.1186/s40359-025-02796-9

Gao, R., Liu, B., Zhao, M., &amp; Zhao, Y. (2025). When and how do antagonistic Machiavellians achieve greater promotability. Personality and Individual Differences, 241, 113180. https://doi.org/10.1016/j.paid.2025.113180

Geis, F. L., &amp; Moon, T. H. (1981). Machiavellianism and deception. Journal of Personality and Social Psychology, 41(4), 766–775. https://doi.org/10.1037/0022-3514.41.4.766

Genau, H. A., Blickle, G., Schütte, N., &amp; Meurs, J. A. (2021). Machiavellian Leader Effectiveness. Journal of Personnel Psychology, 21(1), 1–10. https://doi.org/10.1027/1866-5888/a000284

Grieve, R., &amp; Mahar, D. (2010). The emotional manipulation–psychopathy nexus: Relationships with emotional intelligence, alexithymia and ethical position. Personality and Individual Differences, 48(8), 945–950. https://doi.org/10.1016/j.paid.2010.02.028

Hafeez, S., Qadeer, F., John, A., Iqbal, A., &amp; Sarmad, I. (2024). Mach Mechanics: How Self-Efficacy Drives Machiavelli's Job Performance. Journal of Xi'an Shiyou University, 
Natural Science Edition, 20(7). http://xisdxjxsu.asia/ISSN:1673-064X

Jahangir, M., Shah, S. M., Zhou, J. S., Lang, B., &amp; Wang, X. P. (2025). Machiavellianism: Psychological, Clinical, and Neural Correlations. The Journal of Psychology, 159(3), 155–168. https://doi.org/10.1080/00223980.2024.2382243

Jones, D. N. (2016). The nature of Machiavellianism: Distinct patterns of misbehavior. In 
V. Zeigler-Hill &amp; D. K. Marcus (Eds.), The dark side of personality: Science and practice in social, personality, and clinical psychology (pp. 87–107). American Psychological Association. https://doi.org/10.1037/14854-005

Jones, D. N., &amp; Paulhus, D. L. (2009). Machiavellianism. In M. R. Leary &amp; R. H. Hoyle (Eds.), Handbook of individual differences in social behavior (pp. 93–108). The Guilford Press.

Jones, D. N., &amp; Paulhus, D. L. (2011). The role of impulsivity in the Dark Triad of personality. Personality and Individual Differences, 51(5), 679–682. https://doi.org/10.1016/j.paid.2011.04.011

Kiazad, K., Restubog, S. L. D., Zagenczyk, T. J., Kiewitz, C., &amp; Tang, R. L. (2010). In pursuit of power: The role of authoritarian leadership in the relationship between supervisors’ Machiavellianism and subordinates’ perceptions of abusive supervisory behavior. Journal of Research in Personality, 44(4), 512–519. https://doi.org/10.1016/j.jrp.2010.06.004

Kumar, P. (2025, September 19). ''The Core of Machiavelli’s Political Thought: Power and Pragmatism.'' Political Science Institute. https://polsci.institute/western-political-thought/machiavelli-political-thought-power-pragmatism/

Machiavelli, N. (2017). ''The Prince'' (W. K. Marriott, Trans.). AmazonClassics. (Original work published 1532)

Matthews, M. J., Kelemen, T. K., Matthews, S. H., &amp; Matthews, J. M. (2022). The Machiavellian Organization: A Multilevel Model to Understand Decision Making in Organizations. Group &amp; Organization Management, 47(2), 413–439. https://doi.org/10.1177/10596011221081281

McClelland, D. C. (1965). Toward a theory of motive acquisition. American Psychologist, 20(5), 321–333. https://doi.org/10.1037/h0022225

McHoskey, J. (1995). Narcissism and Machiavellianism. Psychological Reports, 77(3, Pt 1), 755–759. https://doi.org/10.2466/pr0.1995.77.3.755

McHoskey, J. W. (1999). Machiavellianism, Intrinsic Versus Extrinsic Goals, and Social Interest: A Self-Determination Theory Analysis. Motivation and Emotion, 23(4), 267–283. https://doi.org/10.1023/a:1021338809469

Mercadante, E., Aquino, K., Heine, S., &amp; Skarlicki, D. (2026). Why some managers might positively evaluate subordinates with dark personality traits. Journal of Managerial Psychology, 41(3), 458–474. https://doi.org/10.1108/JMP-11-2024-0895.

Moss, J. A., &amp; Barbuto, J. E. (2004). Machiavellianism’s Association with Sources of Motivation and Downward Influence Strategies. Psychological Reports, 94(3), 933–943. https://doi.org/10.2466/pr0.94.3.933-943

MR. BRAIN. (2025, April 22). ''The Dark Truth About Human Nature You Refuse to See {{!}} Machiavelli'' [Video]. Youtube. https://www.youtube.com/watch?v=J9O5lk87z3E 
‌
Paulhus, D. L., &amp; Williams, K. M. (2002). The Dark Triad of personality: Narcissism, Machiavellianism and psychopathy. Journal of Research in Personality, 36(6), 556–563. https://doi.org/10.1016/S0092-6566(02)00505-6

Philosophies for Life. (2024, November 26). ''Niccolò Machiavelli - 6 Powerful Ways to Command Instant Respect (and Never Be Weak Again)'' [Video]. Youtube. https://www.youtube.com/watch?v=w5nJ4XIHv5Q&amp;t=1s 

Rauthmann, J. F. (2012). The Dark Triad and interpersonal perception: Similarities and differences in the social consequences of narcissism, Machiavellianism, and psychopathy. Social Psychological and Personality Science, 3(4), 487–496. https://doi.org/10.1177/1948550611427608

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Spangler, W., &amp; Tikhomirov, A. (2025). Personal power v. socialized power: What Machiavelli and St. Francis can tell us about modern CEOs. https://doi.org/10.64628/aai.h3apd3ujx

Szabó, Z. P., Diller, S. J., Czibor, A., Restás, P., Jonas, E., &amp; Frey, D. (2023). “One of these things is not like the others”: The associations between dark triad personality traits, work attitudes, and work-related motivation. Personality and Individual Differences, 205, 112098. https://doi.org/10.1016/j.paid.2023.112098

Veselka, L., Schermer, J. A., &amp; Vernon, P. A. (2012). The Dark Triad and an expanded framework of personality. Personality and Individual Differences, 53(4), 417–425. https://doi.org/10.1016/j.paid.2012.01.002

Wai, M., &amp; Tiliopoulos, N. (2012). The affective and cognitive empathic nature of the dark triad of personality. Personality and Individual Differences, 52(7), 794–799. https://doi.org/10.1016/j.paid.2012.01.008

White, L. K., Valos, N., Xochitl, &amp; Willis, M. L. (2024). Machiavellianism and Intimate Partner Violence Perpetration: A Systematic Review and Meta-Analysis. Trauma Violence &amp; Abuse, 25(5). https://doi.org/10.1177/15248380241270027

Wills, M. (2021, November). ''Machiavelli, Prince of…Democracy?'' JSTOR Daily. https://daily.jstor.org/machiavelli-prince-of-democracy/

Wilson, D. S., Near, D., &amp; Miller, R. R. (1996). Machiavellianism: A synthesis of the evolutionary and psychological literatures. Psychological Bulletin, 119(2), 285–299. https://doi.org/10.1037/0033-2909.119.2.285 

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Machiavellianism]]</text>
      <sha1>ttawd6dbrcno9xuf2o4zjx9kr0zd8pb</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Mindsets and stigma</title>
    <ns>0</ns>
    <id>331092</id>
    <revision>
      <id>2834776</id>
      <parentid>2834456</parentid>
      <timestamp>2026-09-28T02:00:10Z</timestamp>
      <contributor>
        <username>U3279062</username>
        <id>3106301</id>
      </contributor>
      <comment>light edits on written paragraphs</comment>
      <origin>2834776</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23300" sha1="nfeiwg97vp3638ho2dn32t4o2as3afg" xml:space="preserve">{{title|Mindset and stigma:&lt;br&gt;What role do growth versus fixed mindsets play in prejudice and stigma?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
[[File:Fixed versus growth mindset.png|thumb|Figure 1. Fixed and growth mindsets provide contrasting ways of understanding whether human characteristics can change]]

;Two different reactions

Emma has recently started a new job. During lunch, she mentions that she has previously received treatment for a substance-use disorder. One colleague believes that personal characteristics are largely fixed and assumes that Emma will always be unreliable. Another colleague also wonders why Emma did not change her behaviour sooner.

The colleagues' reactions demonstrate how beliefs about changeability may influence stigma. A fixed mindset may encourage stable stereotypes, whereas a growth mindset may create hope for change. However, believing that change is possible can also increase blame when a person is perceived as responsible for not changing. 

This chapter examines these competing pathways (see Figure 1)
{{RoundBoxBottom}}

* Growth and fixed mindsets describe whether human characteristics are perceived as changeable or relatively permanent (Molden &amp; Dweck, 2006).
* Fixed mindsets may reinforce [[wikipedia:Social_stigma|stigma]] by encouraging essentialist beliefs, stereotypes and pessimism about people's capacity to change (Levy &amp; Dweck, 1999).
* Growth mindsets may weaken essentialism and encourage positive social contract, but they may also increase personal blame when a stigmatised characteristic is considered controllable (Hoyt &amp; Burntt, 2020).
* Understanding these competing effects is important for designing interventions that reduce prejudice without blaming members of stigmatised groups.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What are growth and fixed mindsets, and how are they connected to stigma
* How do fixed mindsets contribute to stretyping, essentialism and prejudice
* Through what processes growth mindsets reduce prejudice and stigma?
* When can growth mindsets reduce prejudice and stigma?
* How can mindset research inform effective stigma-reduction interventions

{{RoundBoxBottom}}

== What are growth and fixed mindsets? ==
Mindsets, also called implicit theories, are beliefs about whether human attributes are fixed or can develop. The word ''implicit'' does not mean that people are necessarily unaware of these beliefs. Rather, it indicates that mindsets function as informal theories through which people organise social information and explain behaviour (Molden &amp; Dweck, 2006). They influence what people notice, the causes they assign to success and failure, and what they expect to happen next.

=== Fixed mindsets ===
A fixed or entity mindset proposes that an attribute reflects an enduring quality. Someone holding a fixed mindset about personality might interpret an unfriendly act as evidence that a person ''is'' unfriendly. A growth or incremental mindset proposes that attributes can develop through learning, experience, strategies, support and changing circumstances. A person with this view may still judge the behaviour negatively, but is more likely to ask what produced it and whether it could change. 

Mindsets are not necessarily global personality traits. The same person may believe that intelligence can develop, but that prejudice is permanent, or that recovery from depression is possible while body weight is entirely controllable. This domain specificity is crucial for understanding stigma. A general endorsement of &quot;growth&quot; does not reliably reveal how someone will judge a particular group or condition.

Mindsets influence motivation because expectations about change affect whether action seems worthwhile. If prejudice is seen as fixed, learning about bias or attempting a difficult intergroup conversation may seem pointless. If it is malleable, effort can seem meaningful. Growth beliefs are generally associated with mastery-oriented goals and adaptive responses to setbacks, although effects vary across contexts (Burnette et al., 2013).

Mindsets also shape attribution, or the causes assigned to behaviour. Entity theorists tend to favour dispositional explanations, inferring stable traits from actions. Chiu et al. (1997) found that implicit theories of personality were connected with this tendency toward lay dispositions. Incremental theorists are comparatively attentive to processes and situations. This does not mean that growth-minded people ignore personal responsibility. Instead, they are more likely to understand behaviour as developing through an interaction among the person, experience and environment.

This distinction connects mindsets with psychological essentialism. Essentialism is the belief that members of a category share a deep, defining essence that produces their observable characteristics. Social categories may consequently appear uniform, natural and sharply separated from one another. A fixed mindset can strengthen essentialism because stable attributes seem to require an enduring internal cause. A growth mindset can weaken it by highlighting variability, development and context. However, fixedness and essentialism are not identical; a characteristic can be stable without being biological, and biological explanations do not always imply absolute permanence. A careful theoretical framework therefore treats mindset, essentialism, controllability and blame as related but distinct constructs.

=== Growth mindsets ===

* A growth or incremental mindset views attributes as capable of development through experience, strategies, support and changes in circumstances.
* Growth mindsets are associated with mastery-orientated goals and adaptive responses to setbacks (Burnette et al., 2013)

== How do fixed mindsets sustain prejudice and stigma? ==
Prejudice refers to negative attitudes toward people because of their group membership, whereas stigma involves social devaluation associated with an attribute, identity or condition. Stigma includes stereotypes, emotional reactions, status loss and discriminatory behaviour. Mindsets contribute to these processes by shaping whether people view differences as defining and whether future improvement seems possible.

=== Essentialism and stereotyping ===
Fixed mindsets encourage observers to convert behaviour into identity. When people treat an action as evidence of an unchanging essence, differences within a group receive less attention, and stereotyping becomes easier. Levy et al. (1998) reported across five experiments that entity theories were associated with greater stereotype formation and endorsement. Levy and Dweck (1999) extended this work to children. Children judged unfamiliar schools described through positive or negative behaviours; those encouraged to adopt static conceptions of people formed stronger stereotypical impressions than those given dynamic conceptions.

These experiments help establish causality because researchers changed the framework through which participants processed group information. Using unfamiliar groups also reduced the influence of established political attitudes. Nevertheless, artificial school and brief messages are not equivalent to historically marginalised groups. Institutions, media and unequal contact reinforce real-world stereotypes. The findings demonstrate a cognitive mechanism, not proof that a short message can undo entrenched stigma.

Essentialism can legitimise inequality by making social arrangements appear inevitable. If poverty is interpreted as evidence of an enduring type of person, structural causes such as labour conditions, housing costs or discrimination become less visible. Similarly, if addiction is regarded as a fixed moral defect, a person's recovery may be discounted, and exclusion may appear sensible. Fixed beliefs can therefore preserve stigma even when overt hostility is socially unacceptable; people may frame avoidance as realism rather than prejudice.

'''&lt;big&gt;Pessimistic expectations and avoidance&lt;/big&gt;'''

Fixed mindsets can also affect intergroup behaviour by producing pessimistic expectations. If people assume that their own bias cannot change, they may anticipate anxiety, failure or appearing prejudiced during an interaction. Avoidance then becomes a protective strategy. Unfortunately, avoiding contact removes opportunities to discover individual variation and experience cooperation, leaving negative expectations untested. 

Carr et al. (2012) examined this process across eight studies. White participants who viewed prejudice as fixed showed less interest in interracial interaction, diversity activities and efforts to reduce their own prejudice. These patterns occurred beyond participants' measured explicit or implicit prejudice. In laboratory interactions, fixed beliefs were associated with less friendly behaviour, even when participants did not report stronger racial prejudice. The research usefully distinguishes hostility from motivation withdrawal; discriminatory social outcomes can arise because a person expects intergroup contact to go badly, not only because they consciously dislike another group.

== How can growth mindsets reduce prejudice and stigma? ==

* Growth mindsets may reduce stigma by challenging the assumption that individuals and groups possess permanent negative qualities. 
* They can promote situational explanations, expectations of improvement and openness to new information.
* Reduced essentialism and more positive intergroup contract are two important stigma-reduction.

=== Reduced essentialism ===

* Growth mindsets weaken assumptions that behaviour reflects an inherent and unchangeable personal or group essence 
* Reduced essentialism may make people less likely to generalise one person's behaviour to an entire social group.
* Growth beliefs encourage greater attention to circumstances, learning and the possbility of individual variation.
* This pathway may reduce prejudice by making stereotypical judgement appear less natural or inevitable (Hoyt &amp; Burnette, 2025).

=== Positive intergroup contact ===

* Growth mindsets weaken assumptions that behaviour reflects an inherent and unchangeable personal or group essence.
* Reduce essentialism may make people less likely to generalise one person's behaviour to an entire social group.
* Growth beliefs encourage greater attention to circumstances, learning and the possibility of individual variation.
* This pathway may reduce intergroup anxiety and increase willingness to communicate.
* Believing that prejudice can change has been associated with improved interracial interactions (Carr et al., 2012).
* Growth mindsets have also been associated with stronger intentions to engage in positive interracial contact (Hoyt et al., 2024)

{{RoundBoxTop|theme=2}}
'''Case study follow-up'''

Emma's colleague learns that recovery is possible but is influenced by treatment access, social support, discrimination and life circumstances. Instead of assuming that Emma is permanately unrealiable or entirely responsible for her past difficulties, the colleague recognises both her capacity for growth and the barriers she has faced.
{{RoundBoxBottom}}

== When can growth mindsets increase stigma? ==

* Growth mindsets are not automatically protective against prejudice because changeability can be interpreted as controllability.
* If a characteristic seems controllable, observes may conclude that the person is responsible for developing or continuing to possess it.
* Consequently, growth beliefs may reduce stigma through lower essentialism while simultaneously increasing it through greater blame.

=== Perceived control and personal blame ===

* Growth messages may imply that people should be capable of changing through sufficient effort.
* If change does not occur, observers may attribute this to laziness, poor choices or a lack of motivation.
* Research on weight stigma found that changeability beliefs could reduce essentialism while also increasing blame (Hoyt et al., 2017).
* This combination is described as a &quot;double-edged sword&quot; because the two pathways can produce opposing effects on prejudice (Hoyt &amp; Burnette, 2020).

=== Structural and situational barriers ===

* Individual effort occurs within social conditions that may either support or restrict opportunities for change.
* Poverty, discrimination, treatment access, social support and institutional policies can influence outcomes.
* Growth messages that ignore these barriers risk presenting complex stigmatised conditions as matters of individual choice.

== How can mindsets inform stigma-reduction interventions? ==

* Mindset research can inform interventions in education, healthcare, workplace and public communication.
* Effective interventions should preserve hope and self-efficacy while avoiding messages that intensify blame.
* Strategies should address social environments and structural barriers in addition to individual beliefs.

=== Designing balanced growth messages ===

* Messages should explain that change may be possible through effective strategies, appropriate resources and social support.
* They should explicitly distinguish between changeability and complete personal controllability. 
* Interventions should acknowledge the environmental and structural barriers that can make change difficult.
* Balanced messages can promote reduced essentialism without implying that people are solely responsible for their stigmatised circumstances (Hoyt &amp; Burnette, 2025).

=== Practical applications ===

* Schools can teach students that stereotypes and prejudiced attitudes can be identified, challenged and changed.
* Healthcare campaigns can communicate hope about treatment and recovery while acknowledging barriers and avoiding blame.
* Workplaces can combine growth-oriented inclusion programs with policies addressing discrimination and unequal opportunities.
* Interventions should be evaluated within the particular population and type of stigma rather than assuming that one message will in every context.

==Figures==
[[File:Share with mental health or development disorder, COD.svg|alt=|thumb|140x140px|'''Figure 2 - The change of population of those with mental illness's over decades''']]
*Three-quarters of Australian men and women survey respondents reported that they would be comfortable talking to their partner about mental health problems (figure 2).  Around one-sixth (16%) were not sure whether they would be comfortable, and a further ten per cent indicated they would not be comfortable [https://www.relationships.org.au/document/october-2015-mental-health-stigma/ (Mental Heath, 2026).]

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Feature box}}
;Feature boxes
;
'''Table 1'''
{| class=&quot;wikitable&quot;
!Aspect
!Fixed mindset
!Growth mindset
|-
|Core belief
|Personal and group characteristics are relatively permanent.
|Personal characteristics and attitudes can develop or change.
|-
|Explanation of behaviour
|Emphasises enduring traits and internal characteristics.
|Considers learning, circumstances, strategies and social support.
|-
|Connection with essentialism
|May strengthen beliefs that groups possess an inherent and defining essence.
|May weaken beliefs that group differences are natural and permanent.
|-
|Effect on stereotypes
|May encourage generalisation and stable stereotypical judgements.
|May encourage attention to individual differences and new information.
|-
|Intergroup expectations
|May create pessimism about whether people, prejudice or relationships can improve.
|May produce more positive expectations and encourage intergroup contact.
|-
|Potential benefit
|May reduce blame when a characteristic is perceived as uncontrollable.
|May reduce prejudice through lower essentialism and greater expectations for change.
|-
|Potential risk
|May promote social avoidance and the belief that stigmatised people cannot change.
|May increase blame when people are considered responsible for not changing.
|}

==Conclusion==

* Fixed mindsets can sustain prejudice by encouraging essentialism, stereotyping and pessimistic expectations about change.
* Growth mindsets can reduce stigma by weakening essentialist beliefs and encouraging positive expectations about individuals and intergroup contact.
* However, growth mindsets may increase stigma when changeability is interrupted as personal controllability and responsibility.
* The role of mindset is therefore contextual rather than universally positive or negative.
* Effective stigma-reduction interventions should promote realistic possibilities for growth while recognising situational and structural barriers.
* Overall, growth mindsets are most likely to reduce prejudice when messages combine hope for change with empathy, social support and reduced blame. 

==See also==

* [[Motivation and emotion/Book/2018/Growth mindset and challenge|Growth mindset and challenge]] (Book chapter, 2018)
* [[wikipedia:Mindset|Mindset]] (Wikipedia)
* [[wikipedia:Psychological essentialism|Psychological essentialism]] (Wikipedia)
* [[wikipedia:Social stigma|Social stigma]] (Wikipedia)

==References==
{{Hanging indent|1=

Burnette, J. L., O'Boyle, E. H., VanEpps, E. M., Pollack, J. M., &amp; Finkel, E. J. (2013). Mind-sets matter: A meta-analytic review of implicit theories and self-regulation. ''Psychological Bulletin'', ''139''(3), 655–701. https://doi.org/10.1037/a0029531

Carr, P. B., Dweck, C. S., &amp; Pauker, K. (2012). “Prejudiced” behavior without prejudice? Beliefs about the malleability of prejudice affect interracial interactions. ''Journal of Personality and Social Psychology'', ''103''(3), 452–471. https://doi.org/10.1037/a0028849

Chiu, C., Hong, Y., &amp; Dweck, C. S. (1997). Lay dispositionism and implicit theories of personality. ''Journal of Personality and Social Psychology'', ''73''(1), 19–30. https://doi.org/10.1037/0022-3514.73.1.19

Hoyt, C. L., &amp; Burnette, J. L. (2020). Growth mindset messaging in stigma-relevant contexts: Harnessing benefits without costs. ''Policy Insights from the Behavioral and Brain Sciences'', ''7''(2), 157–164. https://doi.org/10.1177/2372732220941216

Hoyt, C. L., &amp; Burnette, J. L. (2025). How mindsets can mitigate or sustain prejudice. ''Current Directions in Psychological Science'', ''34''(2), 82–87. https://doi.org/10.1177/09637214241301290

Hoyt, C. L., Burnette, J. L., Auster-Gussman, L., Blodorn, A., &amp; Major, B. (2017). The obesity stigma asymmetry model: The indirect and divergent effects of blame and changeability beliefs on antifat prejudice. ''Stigma and Health'', ''2''(1), 53–65. https://doi.org/10.1037/sah0000026

Hoyt, C. L., Rafferty, D., Earl, S., &amp; Burnette, J. L. (2024). Growth mindsets of people can promote interracial contact intentions among White Americans via positive expectations and reduced anxiety. ''Group Processes &amp; Intergroup Relations'', ''27''(4), 903–924. https://doi.org/10.1177/13684302231187262

Levy, S. R., &amp; Dweck, C. S. (1999). The impact of children's static versus dynamic conceptions of people on stereotype formation. ''Child Development'', ''70''(5), 1163–1180. https://doi.org/10.1111/1467-8624.00085

Molden, D. C., &amp; Dweck, C. S. (2006). Finding “meaning” in psychology: A lay theories approach to self-regulation, social perception, and social development. ''American Psychologist'', ''61''(3), 192–203. https://doi.org/10.1037/0003-066X.61.3.192

Neel, R., &amp; Shapiro, J. R. (2012). Is racial bias malleable? Whites' lay theories of racial bias predict divergent strategies for interracial interactions. ''Journal of Personality and Social Psychology'', ''103''(1), 101–120. https://doi.org/10.1037/a0028237}}

==External links==

* [https://implicit.harvard.edu/implicit/takeatest.html Project Implicit] (Harvard University)
* [https://www.apa.org/topics/racism-bias-discrimination Racism, bias, and discrimination] (American Psychological Association)

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Mindset/Growth]]
[[Category:Motivation and emotion/Book/Stigma]]</text>
      <sha1>nfeiwg97vp3638ho2dn32t4o2as3afg</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Romantic entertainment and love beliefs</title>
    <ns>0</ns>
    <id>331103</id>
    <revision>
      <id>2834795</id>
      <parentid>2834446</parentid>
      <timestamp>2026-09-28T04:47:18Z</timestamp>
      <contributor>
        <username>U3247927</username>
        <id>3005952</id>
      </contributor>
      <comment>editing romantic ideals</comment>
      <origin>2834795</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19991" sha1="b2lzstyonybb5jdhwkelipb71ee3gkb" xml:space="preserve">{{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Couple cuddling on the couch with a movie.png|right|250x250px]]

'''Scenario'''

After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}}  a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that &quot;Romantic movies make relationships looks so effortless. I wish real life worked like that&quot;
{{RoundBoxBottom}}

Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ?    

As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function.

Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner &amp; Wilson, 2013).

Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021).

Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What are romantic beliefs and relationship expectations and why are they important?
* What romantic ideals and relationship messages are commonly portrayed in romantic entertainment?
* How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships?
* What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge?
{{RoundBoxBottom}}

== Romantic beliefs and relationship expectations ==

==== What is love? ====
[[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. It can be thought as triangle vertex with intimacy, desire and decision/ commitment, the intensity of the three points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia &amp; Dunne, 2024)'''.'''''

==== Romantic relationships ====
Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships.  

==== Romantic ideals / romanticism ====
Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher &amp; Metts, 1989; Knox &amp; Sporakowski, 1968, as cited in Sprecher &amp; Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle (&quot;love finds a way&quot;), the existence of a soulmate (&quot;one and only&quot;), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner &amp; Wilson, 2013'').
* '''Love finds a way - belief that love can overcome all obstacles'''
* '''One and only - belief that we have &quot;soulmates&quot;''' 
* '''Idealisation - belief that a &quot;true love&quot; will nearly be perfect'''
* '''Love at first sight - belief when you meet the right person, you will know'''

It is important to note that these beliefs about love represent endorsement of romanticism, not feelings for one person. An individual accepts a set of cultural ideas about how love works and holds these views before or apart from any real romance.

Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014).

== Romantic ideals portrayed in entertainment ==
* Romantic films and romantic comedies
* Television drama
* Reality dating and dating competition shows
* Romance novels
* Popular music
* Social media

Studies investigating romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (''2013'') found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of one's partner. This suggests that the purpose behind media consumption may influence how individuals interpret and apply romantic messages.

Beyond using romantic entertainment as a source of relationship knowledge, individuals may also consume romantic narratives because they become emotionally invested in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances may provide an opportunity for individuals to explore romantic experiences, develop understandings of intimacy and reflect on relationship possibilities without directly experiencing them. Therefore, fictional relationships may serve both an entertainment function and a psychological function by allowing audiences to engage with romantic themes and relationship dynamics.

=== Romantic ideals portrayed in entertainment ===
Discuss things like :

Romantic films / rom coms

* soulmates
* grand gestures
* destiny
* love conquers all
* attractive partners
* conflict resolved quickly
* overcoming obstacles
* happily-ever-after ending

=== Reality dating television ===
* accelerated relationship development
* attraction and compatibility
* dating as entertainment

=== Popular music ===
* love songs reinforce cultural ideas about passion, intimacy and commitment

== Psychological mechanisms of media influence ==
'''Social exchange theory (SET)'''

In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives.

Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships.

=== Romantic scripts and schemas ===

=== Cultivation theory ===

=== Social cognitive theory ===

Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.&lt;quiz display=&quot;bold&quot;&gt;
{A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?:
|type=&quot;()&quot;}
- She has developed a parasocial relationship with the fictional husband
+ She is comparing her relationship to the idealised rewards and expectations portrayed in the media
- She is experiencing attachment insecurity caused by watching romantic television
&lt;/quiz&gt;

== Benefits and drawbacks of romantic entertainment ==

=== Potential benefits ===

=== Potential drawbacks ===

==== Social Exchange Theory and relationship expectations ====
Use this heading structure:
* [[#Overview|Overview]]
*
* What is love
*
*
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner &amp; Kretz, 2021). Following the same beliefs as romantic comedy films.

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]]
* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

=== To what extent should people use romantic entertainment as a guide for real-life relationships? ===
This is essentially your concluding discussion{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33.

Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456.

Chen, Y., Xia, M., &amp; Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442

Hefner, V., &amp; Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290

Hefner, V., &amp; Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697

Lippman, J. R., Ward, L. M., &amp; Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034

Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318

Sprecher, S., &amp; Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009

Van Monsjou, E., &amp; Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Love]]
[[Category:Motivation and emotion/Book/Media]]
[[Category:Motivation and emotion/Book/Relationships]]</text>
      <sha1>b2lzstyonybb5jdhwkelipb71ee3gkb</sha1>
    </revision>
    <revision>
      <id>2834802</id>
      <parentid>2834795</parentid>
      <timestamp>2026-09-28T05:05:15Z</timestamp>
      <contributor>
        <username>U3247927</username>
        <id>3005952</id>
      </contributor>
      <comment>/* Romantic ideals / romanticism */</comment>
      <origin>2834802</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20060" sha1="5hgyulayf2we3jzll47jq1vb6mecnx3" xml:space="preserve">{{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Couple cuddling on the couch with a movie.png|right|250x250px]]

'''Scenario'''

After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}}  a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that &quot;Romantic movies make relationships looks so effortless. I wish real life worked like that&quot;
{{RoundBoxBottom}}

Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ?    

As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function.

Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner &amp; Wilson, 2013).

Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021).

Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What are romantic beliefs and relationship expectations and why are they important?
* What romantic ideals and relationship messages are commonly portrayed in romantic entertainment?
* How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships?
* What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge?
{{RoundBoxBottom}}

== Romantic beliefs and relationship expectations ==

==== What is love? ====
[[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. It can be thought as triangle vertex with intimacy, desire and decision/ commitment, the intensity of the three points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia &amp; Dunne, 2024)'''.'''''

==== Romantic relationships ====
Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships.  

==== Romantic ideals / romanticism ====
Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher &amp; Metts, 1989; Knox &amp; Sporakowski, 1968, as cited in Sprecher &amp; Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle (&quot;love finds a way&quot;), the existence of a soulmate (&quot;one and only&quot;), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner &amp; Wilson, 2013'').
* '''Love finds a way - belief that love can overcome all obstacles'''
* '''One and only - belief that we have &quot;soulmates&quot;''' 
* '''Idealisation - belief that a &quot;true love&quot; will nearly be perfect'''
* '''Love at first sight - belief when you meet the right person, you will know'''

Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent  shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher &amp; Metts, 1989). 

== Romantic ideals portrayed in entertainment ==
* Romantic films and romantic comedies
* Television drama
* Reality dating and dating competition shows
* Romance novels
* Popular music
* Social media

Studies investigating romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (''2013'') found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of one's partner. This suggests that the purpose behind media consumption may influence how individuals interpret and apply romantic messages.

Beyond using romantic entertainment as a source of relationship knowledge, individuals may also consume romantic narratives because they become emotionally invested in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances may provide an opportunity for individuals to explore romantic experiences, develop understandings of intimacy and reflect on relationship possibilities without directly experiencing them. Therefore, fictional relationships may serve both an entertainment function and a psychological function by allowing audiences to engage with romantic themes and relationship dynamics.

=== Romantic ideals portrayed in entertainment ===
Discuss things like :

Romantic films / rom coms

* soulmates
* grand gestures
* destiny
* love conquers all
* attractive partners
* conflict resolved quickly
* overcoming obstacles
* happily-ever-after ending

=== Reality dating television ===
* accelerated relationship development
* attraction and compatibility
* dating as entertainment

=== Popular music ===
* love songs reinforce cultural ideas about passion, intimacy and commitment

== Psychological mechanisms of media influence ==
'''Social exchange theory (SET)'''

In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives.

Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships.

=== Romantic scripts and schemas ===

=== Cultivation theory ===

=== Social cognitive theory ===

Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.&lt;quiz display=&quot;bold&quot;&gt;
{A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?:
|type=&quot;()&quot;}
- She has developed a parasocial relationship with the fictional husband
+ She is comparing her relationship to the idealised rewards and expectations portrayed in the media
- She is experiencing attachment insecurity caused by watching romantic television
&lt;/quiz&gt;

== Benefits and drawbacks of romantic entertainment ==
Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014).

=== Potential benefits ===

=== Potential drawbacks ===

==== Social Exchange Theory and relationship expectations ====
Use this heading structure:
* [[#Overview|Overview]]
*
* What is love
*
*
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner &amp; Kretz, 2021). Following the same beliefs as romantic comedy films.

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]]
* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

=== To what extent should people use romantic entertainment as a guide for real-life relationships? ===
This is essentially your concluding discussion{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33.

Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456.

Chen, Y., Xia, M., &amp; Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442

Hefner, V., &amp; Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290

Hefner, V., &amp; Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697

Lippman, J. R., Ward, L. M., &amp; Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034

Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318

Sprecher, S., &amp; Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009

Van Monsjou, E., &amp; Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Love]]
[[Category:Motivation and emotion/Book/Media]]
[[Category:Motivation and emotion/Book/Relationships]]</text>
      <sha1>5hgyulayf2we3jzll47jq1vb6mecnx3</sha1>
    </revision>
    <revision>
      <id>2834806</id>
      <parentid>2834802</parentid>
      <timestamp>2026-09-28T05:25:37Z</timestamp>
      <contributor>
        <username>U3247927</username>
        <id>3005952</id>
      </contributor>
      <comment>intro to entertainment media</comment>
      <origin>2834806</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20337" sha1="9bcqlepl100km7iz4xaqbl7cgm0sgak" xml:space="preserve">{{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Couple cuddling on the couch with a movie.png|right|250x250px]]

'''Scenario'''

After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}}  a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that &quot;Romantic movies make relationships looks so effortless. I wish real life worked like that&quot;
{{RoundBoxBottom}}

Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ?    

As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function.

Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner &amp; Wilson, 2013).

Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021).

Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What are romantic beliefs and relationship expectations and why are they important?
* What romantic ideals and relationship messages are commonly portrayed in romantic entertainment?
* How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships?
* What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge?
{{RoundBoxBottom}}

== Romantic beliefs and relationship expectations ==

==== What is love? ====
[[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. It can be thought as triangle vertex with intimacy, desire and decision/ commitment, the intensity of the three points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia &amp; Dunne, 2024)'''.'''''

==== Romantic relationships ====
Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships.  

==== Romantic ideals / romanticism ====
Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher &amp; Metts, 1989; Knox &amp; Sporakowski, 1968, as cited in Sprecher &amp; Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle (&quot;love finds a way&quot;), the existence of a soulmate (&quot;one and only&quot;), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner &amp; Wilson, 2013'').
* '''Love finds a way - belief that love can overcome all obstacles'''
* '''One and only - belief that we have &quot;soulmates&quot;''' 
* '''Idealisation - belief that a &quot;true love&quot; will nearly be perfect'''
* '''Love at first sight - belief when you meet the right person, you will know'''

Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent  shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher &amp; Metts, 1989). 

== Romantic ideals portrayed in entertainment ==
* Romantic films and romantic comedies
* Television drama
* Reality dating and dating competition shows
* Romance novels
* Popular music
* Social media

Studies investigating romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (''2013'') found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of one's partner. This suggests that the purpose behind media consumption may influence how individuals interpret and apply romantic messages.

Beyond using romantic entertainment as a source of relationship knowledge, individuals may also consume romantic narratives because they become emotionally invested in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances may provide an opportunity for individuals to explore romantic experiences, develop understandings of intimacy and reflect on relationship possibilities without directly experiencing them. Therefore, fictional relationships may serve both an entertainment function and a psychological function by allowing audiences to engage with romantic themes and relationship dynamics.

=== Romantic ideals portrayed in entertainment ===
Romantic ideals are influenced by a range of social and cultural factors; however these beliefs do not develop in isolation. Entertainment media play a significant in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy &amp; relationship development.

Romantic films / rom coms

* soulmates
* grand gestures
* destiny
* love conquers all
* attractive partners
* conflict resolved quickly
* overcoming obstacles
* happily-ever-after ending

=== Reality dating television ===
* accelerated relationship development
* attraction and compatibility
* dating as entertainment

=== Popular music ===
* love songs reinforce cultural ideas about passion, intimacy and commitment

== Psychological mechanisms of media influence ==
'''Social exchange theory (SET)'''

In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives.

Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships.

=== Romantic scripts and schemas ===

=== Cultivation theory ===

=== Social cognitive theory ===

Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.&lt;quiz display=&quot;bold&quot;&gt;
{A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?:
|type=&quot;()&quot;}
- She has developed a parasocial relationship with the fictional husband
+ She is comparing her relationship to the idealised rewards and expectations portrayed in the media
- She is experiencing attachment insecurity caused by watching romantic television
&lt;/quiz&gt;

== Benefits and drawbacks of romantic entertainment ==
Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014).

=== Potential benefits ===

=== Potential drawbacks ===

==== Social Exchange Theory and relationship expectations ====
Use this heading structure:
* [[#Overview|Overview]]
*
* What is love
*
*
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner &amp; Kretz, 2021). Following the same beliefs as romantic comedy films.

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]]
* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

=== To what extent should people use romantic entertainment as a guide for real-life relationships? ===
This is essentially your concluding discussion{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33.

Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456.

Chen, Y., Xia, M., &amp; Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442

Hefner, V., &amp; Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290

Hefner, V., &amp; Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697

Lippman, J. R., Ward, L. M., &amp; Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034

Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318

Sprecher, S., &amp; Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009

Van Monsjou, E., &amp; Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Love]]
[[Category:Motivation and emotion/Book/Media]]
[[Category:Motivation and emotion/Book/Relationships]]</text>
      <sha1>9bcqlepl100km7iz4xaqbl7cgm0sgak</sha1>
    </revision>
    <revision>
      <id>2834807</id>
      <parentid>2834806</parentid>
      <timestamp>2026-09-28T05:42:14Z</timestamp>
      <contributor>
        <username>U3247927</username>
        <id>3005952</id>
      </contributor>
      <comment>/* Romantic ideals portrayed in entertainment */</comment>
      <origin>2834807</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20309" sha1="10r99n7inx8brv6xg80nqwl5ts3oi11" xml:space="preserve">{{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Couple cuddling on the couch with a movie.png|right|250x250px]]

'''Scenario'''

After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}}  a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that &quot;Romantic movies make relationships looks so effortless. I wish real life worked like that&quot;
{{RoundBoxBottom}}

Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ?    

As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function.

Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner &amp; Wilson, 2013).

Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021).

Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What are romantic beliefs and relationship expectations and why are they important?
* What romantic ideals and relationship messages are commonly portrayed in romantic entertainment?
* How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships?
* What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge?
{{RoundBoxBottom}}

== Romantic beliefs and relationship expectations ==

==== What is love? ====
[[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. It can be thought as triangle vertex with intimacy, desire and decision/ commitment, the intensity of the three points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia &amp; Dunne, 2024)'''.'''''

==== Romantic relationships ====
Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships.  

==== Romantic ideals / romanticism ====
Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher &amp; Metts, 1989; Knox &amp; Sporakowski, 1968, as cited in Sprecher &amp; Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle (&quot;love finds a way&quot;), the existence of a soulmate (&quot;one and only&quot;), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner &amp; Wilson, 2013'').
* '''Love finds a way - belief that love can overcome all obstacles'''
* '''One and only - belief that we have &quot;soulmates&quot;''' 
* '''Idealisation - belief that a &quot;true love&quot; will nearly be perfect'''
* '''Love at first sight - belief when you meet the right person, you will know'''

Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent  shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher &amp; Metts, 1989). 

== Romantic ideals portrayed in entertainment ==
Romantic ideals are influenced by a range of social and cultural factors; however, these beliefs do not develop in isolation. Entertainment media play a significant role in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy, and relationship development. Studies examining romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (2013) found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of romantic partners. Furthermore, romantic comedies frequently depict themes associated with romantic ideals, including love at first sight, soulmates, idealised partners, and the belief that love can overcome obstacles. These findings suggest that media consumption may influence how individuals interpret and apply romantic messages to their own relationships.

Beyond serving as a source of relationship knowledge, romantic entertainment may also attract audiences through emotional investment in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances provides opportunities for individuals to explore romantic experiences, develop understandings of intimacy, and reflect on relationship possibilities without directly experiencing them. Consequently, fictional relationships may serve both entertainment and psychological functions by allowing audiences to engage with romantic themes and relationship dynamics.

Romantic films / rom coms

* soulmates
* grand gestures
* destiny
* love conquers all
* attractive partners
* conflict resolved quickly
* overcoming obstacles
* happily-ever-after ending

=== Reality dating television ===
* accelerated relationship development
* attraction and compatibility
* dating as entertainment

=== Popular music ===
* love songs reinforce cultural ideas about passion, intimacy and commitment

== Psychological mechanisms of media influence ==
'''Social exchange theory (SET)'''

In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives.

Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships.

=== Romantic scripts and schemas ===

=== Cultivation theory ===

=== Social cognitive theory ===

Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.&lt;quiz display=&quot;bold&quot;&gt;
{A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?:
|type=&quot;()&quot;}
- She has developed a parasocial relationship with the fictional husband
+ She is comparing her relationship to the idealised rewards and expectations portrayed in the media
- She is experiencing attachment insecurity caused by watching romantic television
&lt;/quiz&gt;

== Benefits and drawbacks of romantic entertainment ==
Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014).

=== Potential benefits ===

=== Potential drawbacks ===

==== Social Exchange Theory and relationship expectations ====
Use this heading structure:
* [[#Overview|Overview]]
*
* What is love
*
*
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner &amp; Kretz, 2021). Following the same beliefs as romantic comedy films.

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]]
* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

=== To what extent should people use romantic entertainment as a guide for real-life relationships? ===
This is essentially your concluding discussion{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33.

Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456.

Chen, Y., Xia, M., &amp; Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442

Hefner, V., &amp; Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290

Hefner, V., &amp; Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697

Lippman, J. R., Ward, L. M., &amp; Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034

Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318

Sprecher, S., &amp; Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009

Van Monsjou, E., &amp; Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Love]]
[[Category:Motivation and emotion/Book/Media]]
[[Category:Motivation and emotion/Book/Relationships]]</text>
      <sha1>10r99n7inx8brv6xg80nqwl5ts3oi11</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Adaptive versus maladaptive self-reflection</title>
    <ns>0</ns>
    <id>331109</id>
    <revision>
      <id>2834800</id>
      <parentid>2832135</parentid>
      <timestamp>2026-09-28T05:02:54Z</timestamp>
      <contributor>
        <username>U3211150</username>
        <id>3106672</id>
      </contributor>
      <comment>Condense Overview and merge final key points - add external links</comment>
      <origin>2834800</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25468" sha1="infp2oufz70lm3x0khgwpr1918ckz0a" xml:space="preserve">{{title|Adaptive versus maladaptive self-reflection:&lt;br&gt;When does self-reflection promote wellbeing and when does it contribute to psychological distress?}}
__TOC__

==Overview ==
{{RoundBoxTop|theme=3}}

[[File:Le Penseur in the Jardin du Musée Rodin, Paris 14 June 2015.jpg|Le_Penseur_in_the_Jardin_du_Musée_Rodin,_Paris_14_June_2015]|right|thumb|150px|'''Figure 1'''. Reflection may produce insight, but repeated self-focused thought may contribute to distress.]]

'''Scenario: An embarrassing encounter'''

Trevor's day at university was fairly ordinary. While walking to get his second coffee, he exchanged a brief smile with another student. Distracted, he failed to notice the uneven ground and tripped in front of her. He quickly walked away, flushed and deeply embarrassed. Trevor repeatedly replayed the incident in his mind, becoming absorbed in self-focused thought (see Figure 1). He decided that if he saw her again, he would avoid her to prevent further embarrassment. 

When Trevor told his friend Alucard what had happened, Alucard suggested that she might not have noticed. This prompted Trevor to consider whether his prediction of rejection was supported by what had actually occurred. Soon afterwards, Trevor saw the same student approaching. Although he still felt embarrassed, he reconsidered his interpretation and introduced himself. He discovered that she was happy to speak with him.

Trevor's response illustrates how reflecting on the same experience may be associated with different emotional and behavioural outcomes. Why might repeatedly replaying the event have prolonged his embarrassment, while reconsidering his interpretation helped him respond more constructively?
{{RoundBoxBottom}}

[[w:Self-reflection|Self-reflection]] involves examining one’s thoughts, emotions and behaviour, but this process does not necessarily produce insight or psychological wellbeing. Self-reflection and insight are related but distinguishable: reflection describes the process of self-examination, whereas insight involves developing clearer self-understanding (Grant et al., 2002). This distinction is practically important because deliberate self-analysis may appear constructive even when it maintains embarrassment, avoidance or distress.

The effects of self-focused thought may depend on how it is performed. Trapnell and Campbell (1999) distinguished curiosity-driven reflection from recurrent [[w:Rumination (psychology)|rumination]] prompted by perceived threats, losses or injustices involving the self. More broadly, repetitive thought may be relatively constructive or unconstructive depending on its content, context and level of processing (Watkins, 2008). Psychological perspective may also matter, as analysing a negative experience from a self-distanced perspective may reduce immediate emotional reactivity compared with self-immersed analysis (Ayduk &amp; Kross, 2008). Understanding these processes may help explain why reflection sometimes supports insight and constructive action but, in other circumstances, contributes to continuing distress.{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What distinguishes adaptive self-reflection from maladaptive self-focused thought? 
* Under what conditions may self-reflection support psychological wellbeing?
* Under what conditions may self-focused thought contribute to psychological distress?
* What psychological processes influence the outcomes of self-reflection? 
* How might people engage in more adaptive self-reflection?

{{RoundBoxBottom}}

== Understanding adaptive and maladaptive self-reflection ==
'''focus question''' - What distinguishes adaptive self-reflection from maladaptive self-focused thought?

'''Topic sentence/introductory message &lt;- &lt;u&gt;''these subheadings will be used for my planning/paragraph structure and will not be used as  official subheadings for the final book chapter (some may be used if appropriate but not all)''&lt;/u&gt;'''

*Self-focused thought is not explicitly helpful or harmful 
Self-focused thought may involve distinguishable processes which influence the wellbeing outcomes. Whether adaptive or maladaptive behaviours are executed may rely on motivation, processing style, capacity to generate insight and relationship with repetitive negative thinking.

'''Self-reflection definition'''

* Self-reflection can be defined as the inspection and evaluation of one's thoughts, emotions and behaviour. (Grant et al., 2002).
* The act of self-reflection involves directed attention towards internal experiences and processes (Grant et al., 2002)
* Engaging in the act of self-reflection is not the same as gaining understanding of what the internal experiences mean (Grant et al., 2002)
* Adaptive self-reflection may involve the flexible, purposeful processing that supports insight, problem-solving or movement towards personally relevant goals (Watkins, 2008)
* Maladaptive self-reflection may include rumination, which involves a more self-focused thought process that refers to repetitive attention to distress and its causes or consequences, potentially interfering with problem-solving and supportive behaviours. (Nolen-Hoeksema et al., 2008).

'''Self-reflection vs insight'''
* Insight can be defined as the clarity of understanding of one's thoughts, emotions and behaviour. (Grant et al., 2002)
* While self-reflection and insight are related, reflection involves the process of self-examination, whereas insight describes the possible outcome. (Grant et al., 2002)
* Every process is different according to the form, context and individual differences. This implies that an individual may spend a considerable time reflecting but not gain any insight. (Grant et al., 2002)
* When these theories are applied to Trevor's scenario, He replays the incident with self-focused attention, but it does not initially provide him with easing distress or more accurate understanding.

Grant et al. (2002) recognised self-reflection and insight as distinguishable factors. Suggesting that higher levels of self-examination do not with certainty produce clear self-understanding. Results found that self-reflection associated positively with anxiety and stress. Whereas, insight was negatively associated with depression, anxiety and stress, while positively associated with cognitive flexibility and self-regulation. These findings were primarily correlational and participants included a sample of undergraduate psychology students which may limit the generalisability. Therefore, these findings demonstrate association among self-reflection insight and psychological outcomes, but they do not establish causal relationships.  

'''Reflection &amp; Rumination''' 

* The &quot;self-absorption paradox&quot; highlights how self-focused attention can be associated with both self-understanding and psychological distress (Trapnell &amp; Campbell, 1999).
* Trapnell and Campbell (1999) identified two motivationally distinct forms of private self-attention. They distinguish between curiosity-driven reflection or intellectual interest in the self compared to recurrent self-rumination likely to be prompted by threats, losses or injustices regarding the self.
* This distinction suggests that the motivation and quality of self-focused attention may be more informative than the amount of reflection alone
* In relation to the Trevor scenario, the act of repeated replaying the embarrassment and anticipated rejection resembles the threat-focused rumination, whereas questioning his initial assumptions illustrates curiosity-driven reflection.
* Trapnell and Campbell's (1999) model provides a key theoretical differentiation for dispositional reflection and rumination. While their research support the distinction between the different processes of reflection. However, it does not establish causation between reflection supporting adaptive outcomes and repetitive thought causing harmful outcomes.

'''Rumination is not one uniform process'''

* According to Treynor et al. (2003), rumination may contain at least two distinguishable components: brooding and reflective pondering.
* Brooding can be defined as the passive comparison between one's current circumstance and an unattained standard. (Treynor et al., 2003).
* Reflective pondering can be defined as purposeful attention intended to understand or address distress. (Treynor et al., 2003).
* It important to highlight these as differing constructs as they both associate differently when measured alongside depressive symptoms. (Treynor et al., 2003).
* Reflective pondering showed a different and less consistently maladaptive pattern than brooding. Moreover, it is also measured alongside depressive rumination and should not automatically be equated to adaptive self-reflection. This highlighted how different processes of rumination have different relationships with depressive symptoms. (Treynor et al., 2003).
* Important distinction - reflection and reflective pondering are from two different measures and should not be treated as identical constructs

'''Reflection and rumination overlap'''

* Takano and Tanno (2009), proposed that self-reflection and self-rumination are distinguishable but may not be entirely independent.
* A two wave longitudinal study over three weeks reported (Takano and Tanno (2009)):
** Self-reflection was associated with lower depression, whereas self-rumination was associated with higher depression
** Important to note that the estimated total effect of reflection on depression was close to zero, as reflection and rumination occurred together.
** This suggests that adaptive reflection processes may coexist with maladaptive processes that counteract positive outcomes
** The small, predominantly male undergraduate sample and brief follow-up period limit generalisability, while the observational design prevents firm causal conclusions.

'''Concluding sentence''' - research suggests that adaptive self-reflection is more likely to involve curiosity-driven and flexible examination that may facilitate insight. In contrast, maladaptive self-focused thought may involve threat-focused repetitions, passive brooding and difficulty disengaging. However, these processes may interact and fluctuate, so reflection should not be assumed to be beneficial merely because it is deliberate. its outcomes may depend on whether it generates clearer understanding or becomes incorporated into repetitive rumination.

== Adaptive self-reflection and wellbeing ==
'''Focus question -'''  Under what conditions may self reflection support psychological wellbeing?

'''Topic sentence/introductory message -''' Self reflection is more likely to support wellbeing when insight is gained to facilitate constructive meaning,

'''Key points'''

* Insight was positively associated with happiness and life satisfaction, whereas self-reflection alone did not show the same relationship with subjective wellbeing (Lyke, 2009).
* Adaptive self-reflection was indirectly associated with greater resilience and wellbeing through insight when rumination was low. When rumination was high, the pattern was less favourable (Bucknell et al., 2022).
* A randomised control trial, found that self reflective writing was more effective than descriptive writing in reinforced perceived resilience. Additionally, reflection on successful coping demonstrates better maintenance of benefits than reflecting on unsuccessful coping (Bucknell et al., 2024).
* An app-based journal that targeted positive processes was associated with increased psychological wellbeing among participants with average or higher baseline dispositional self-reflection, after statistically controlling for rumination (MacIsaac et al., 2023)

'''Concluding sentence''' - the evidence highlighted that the possible benefits of reflection depend on whether the action produces greater insight, constructing meaning and recognising effective coping. 

Find research potential locus of control x self-reflection as a point 

'''Limitations'''

* Lyke (2009) is correlational and does not prove insight causes wellbeing
* Bucknell et al. (2022) is a cross-sectional moderated mediation analysis, proposed indirect statistical pathway does not infer causation
* Bucknell et al. (2024) experimental outcomes were targeted to perceived resilience rather than self reflection. Participants also consisted of Australian ministry workers limiting generalisability
* MacIsaac et al. (2023), study was longitudinal but lacked experimental assignment and a control group. Participants chose how much they used the journal, and the app contained other features. Additionally, the nature of the quasi-experiment makes it difficult to confirm whether outcomes are influenced by individuals pre existing reflection intentions and behaviours. 

== Maladaptive self-reflection and psychological distress ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]'''Focus questions -''' Under what conditions may self-focused thought contribute to psychological distress?
'''Topic sentence/introductory message -''' Self-focused thought may contribute to psychological distress when it becomes repetitive, negative and difficult to detach from, especially when interfering with constructive behaviour and problem solving.

'''Key points'''

*Response style theory proposes that repetitive focus on distress and its causes and consequences may reinforce negative thinking, impair problem solving and interfere with constructive understanding and behaviours (Nolen-Hoeksema et al., 2008)
*A meta-analysis of effect sizes highlighted an association between self-focused attention and negative affect. The strength of this association differed by the form of self-focus, distress and samples (Mor &amp; Winquist, 2002)
*Cross-sectional mediation models indicated that brooding statistically accounted for part of the relationship between lower self-compassion and depressive symptoms, while brooding and worry correlated with anxiety (Raes, 2010).
*Dunn &amp; Luchner (2022) researched reflection on personal mistakes and how rumination conditions effected negative affect. Results found that no significant improvement in negative emotion, which indicates that rumination may maintain psychological distress.

'''Concluding sentence''' - self-focused thought may involve distress when it maintains attention toward negative content without facilitating insight or effective problem-solving.

'''Limitations'''

* Identifies overall association, not the causality of self-focus and distress. Additionally this meta-analysis is quite dated , social norms and stressors have changed so a more recent meta-analysis may be more useful. (Mor &amp; Winquist, 2002)
* Nolen-Hoeksema et al. (2008) concluded that rumination predicted the onset of depression more consistently than the duration of the depressive episode, findings concerning duration were mixed.
* Raes (2010) cross-sectional design may limit generalisability and causality. 
* Dunn and Luchner (2022) did not find a strong implication that rumination significantly caused a decreased mood. Additionally, the study focuses on reflection of personal mistakes and not self-reflection in general.

== Factors influencing the outcomes of self-reflection ==
'''Focus question -''' What psychological processes influence the outcomes of self-reflection?

'''Topic sentence -''' The outcomes of self-reflection may be influenced by the interaction of cognitive, emotional, personal  and cultural factors rather than on the amount of maladaptive process alone.

'''Key points''' 
*A theoretical review proposed that repetitive thought may be constructive or unconstructive depending on its emotional content, situational context and level of construal, including whether processing is relatively abstract or concrete (Watkins, 2008).
*Experimental evidence suggests that analysing a negative experience from a psychologically distanced perspective may support faster cardiovascular recovery than analysing it from a self-immersed perspective (Ayduk &amp; Kross, 2008).
*Cross-cultural studies found that Russian participants reported greater spontaneous self-distancing and less distress while reflecting on negative experiences than American participants, indicating that cultural context may influence how reflection is performed and experienced (Grossmann &amp; Kross, 2010).
*The relationship between adaptive reflection, insight and wellbeing differed according to participants’ levels of rumination, suggesting that rumination may interfere with otherwise constructive reflective processing (Bucknell et al., 2022).
*An experiment involving reflection on a significant mistake suggested that self-criticism may also influence outcomes, as reflection was associated with reduced negative emotion among participants high in the “hated self” form of self-criticism (Dunn &amp; Luchner, 2022).

'''Concluding sentence''' - important influences may include processing style, psychological distance, emotional content, rumination, self-criticism, culture and the extent to which reflection generates insight or action.

'''Limitations''' 

== Developing more adaptive self-reflection ==

'''Focus question''' - How might people engage in more adaptive self-reflection?

*Adaptive self-reflection may be encouraged by giving reflection a clear structure, directing attention towards specific experiences and supporting perspective, insight and constructive action.
*Analysing a negative experience from a self-distanced perspective may reduce immediate physiological reactivity compared with analysing the experience from a self-immersed perspective (Ayduk &amp; Kross, 2008).
*In a one-month randomised trial with distressed students, both [[w:Mindfulness|mindfulness meditation]] and relaxation training reduced distress and increased positive states of mind relative to a control condition; mindfulness meditation also reduced rumination relative to the control condition (Jain et al., 2007).
*Positively focused journalling prompts concerning growth, mastery, goals and the future were associated with improved psychological wellbeing among participants with average or higher dispositional self-reflection (MacIsaac et al., 2023).
*Structured reflection on successful coping experiences may be particularly useful, as its effects on perceived resilience were maintained more effectively than reflection focused on unsuccessful coping experiences (Bucknell et al., 2024).
*in relation to recalling a significant personal mistake, reflection was associated with reduced negative emotion among participants high in hated-self criticism, whereas rumination produced no significant emotional change. (Dunn &amp; Luchner, 2022).

'''Concluding sentence''' - promising strategies include psychological distancing, mindfulness, positively focused journalling and structured reflection on successful coping; however, these approaches should not be presented as universally effective.

'''Limitations''' 
*Ayduk and Kross (2008), dated source and supports immediate self-distancing over a more universal and long term intervention.
*Jain et al. (2007), Sample included 83 distressed students limiting generalisability. Additionally, mindfulness was not significantly stronger for relaxation for distress or positive states of mind.
*Bucknell et al. (2024), study targeted perceived resilience and was a specific occupational sample.

;Quiz

Time to quiz your knowledge!

Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Self-reflection involves examining one's thoughts, emotions and behaviour:
|type=&quot;()&quot;}
+ True
- False

{Self-reflection is always adaptive when it is deliberate and purposeful:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
*Self-reflection does not appear to be definitively adaptive or maladaptive; its outcomes may depend on how reflection is processed and the context. 
*Adaptive self-reflection may involve curiosity, flexible thinking and movement towards insight, problem-solving or constructive action. 
*Maladaptive self-focused thought may involve threat-focused repetition, passive brooding and difficulty disengaging from distress. 
*Reflection alone does not necessarily produce insight or psychological wellbeing. 
*Factors such as psychological distance, rumination, emotional content, self-criticism and cultural context may influence reflective outcomes. 
*Strategies such as mindfulness, self-distancing, positively focused journalling and structured reflection on successful coping may encourage more adaptive reflection. 

'''Concluding sentence''' - outcomes do not rely on whether an individual reflects or not, but whether their reflection generates clearer understanding and constructive action or maintains repetitive negative attention.

==See also==
* [[w:Emotion regulation|Emotion regulation]] (Wikipedia)
* [[w:Rumination (psychology)|Rumination]] (Wikipedia)
* [[Motivation and emotion/Book/2013/Rumination|Rumination]] (Book chapter, 2013)
* [[w:Self-awareness|Self-awareness]] (Wikipedia)

==References==

{{Hanging indent|1=

Ayduk, Ö., &amp; Kross, E. (2008). Enhancing the pace of recovery: Self-distanced analysis of negative experiences reduces blood pressure reactivity. ''Psychological Science, 19''(3), 229–231. https://doi.org/10.1111/j.1467-9280.2008.02073.x 

Bucknell, K. J., Kangas, M., &amp; Crane, M. F. (2022). Adaptive self-reflection and resilience: The moderating effects of rumination on insight as a mediator. ''Personality and Individual Differences, 185'', 111234. https://doi.org/10.1016/j.paid.2021.111234

Bucknell, K. J., Kangas, M., Karin, E., &amp; Crane, M. F. (2024). A randomized controlled trial comparing the effects of self‐reflective writing focused on successful and unsuccessful coping experiences on resilience. ''Stress and Health, 40''(2), e3311. https://doi.org/10.1002/smi.3311

Dunn, N. A., &amp; Luchner, A. F. (2022). The emotional impact of self‐criticism on self‐reflection and rumination. ''Psychology and Psychotherapy: Theory, Research and Practice, 95''(4), 1126–1139.  https://doi.org/10.1111/papt.12422

Grant, A. M., Franklin, J., &amp; Langford, P. (2002). The self-reflection and insight scale: A new measure of private self-consciousness. ''Social Behavior and Personality: An International Journal, 30''(8), 821–836. https://doi.org/10.2224/sbp.2002.30.8.821

Grossmann, I., &amp; Kross, E. (2010). The impact of culture on adaptive versus maladaptive self-reflection. ''Psychological Science, 21''(8), 1150–1157. https://doi.org/10.1177/0956797610376655

Jain, S., Shapiro, S. L., Swanick, S., Roesch, S. C., Mills, P. J., Bell, I., &amp; Schwartz, G. E. (2007). A randomized controlled trial of mindfulness meditation versus relaxation training: Effects on distress, positive states of mind, rumination, and distraction. ''Annals of Behavioral Medicine, 33''(1), 11–21. https://doi.org/10.1207/s15324796abm3301_2 

Lyke, J. A. (2009). Insight, but not self-reflection, is related to subjective well-being. ''Personality and Individual Differences, 46''(1), 66–70. https://doi.org/10.1016/j.paid.2008.09.010

MacIsaac, A., Mushquash, A. R., &amp; Wekerle, C. (2023). Writing yourself well: Dispositional self-reflection moderates the effect of a smartphone app-based journaling intervention on psychological wellbeing across time. ''Behaviour Change, 40''(4), 297–313. https://doi.org/10.1017/bec.2022.24

Mor, N., &amp; Winquist, J. (2002). Self-focused attention and negative affect: A meta-analysis. ''Psychological Bulletin, 128''(4), 638–662. https://doi.org/10.1037/0033-2909.128.4.638

Nolen-Hoeksema, S., Wisco, B. E., &amp; Lyubomirsky, S. (2008). Rethinking rumination. ''Perspectives on Psychological Science, 3''(5), 400–424. https://doi.org/10.1111/j.1745-6924.2008.00088.x

Raes, F. (2010). Rumination and worry as mediators of the relationship between self-compassion and depression and anxiety. ''Personality and Individual Differences, 48''(6), 757–761. https://doi.org/10.1016/j.paid.2010.01.023

Takano, K., &amp; Tanno, Y. (2009). Self-rumination, self-reflection, and depression: Self-rumination counteracts the adaptive effect of self-reflection. ''Behaviour Research and Therapy, 47''(3), 260–264. https://doi.org/10.1016/j.brat.2008.12.008

Trapnell, P. D., &amp; Campbell, J. D. (1999). Private self-consciousness and the five-factor model of personality: Distinguishing rumination from reflection. ''Journal of Personality and Social Psychology, 76''(2), 284–304. https://doi.org/10.1037/0022-3514.76.2.284

Treynor, W., Gonzalez, R., &amp; Nolen-Hoeksema, S. (2003). Rumination reconsidered: A psychometric analysis. ''Cognitive Therapy and Research, 27''(3), 247–259. https://doi.org/10.1023/A:1023910315561

Watkins, E. (2004). Adaptive and maladaptive ruminative self-focus during emotional processing. ''Behaviour Research and Therapy, 42''(9), 1037–1052. https://doi.org/10.1016/j.brat.2004.01.009

Watkins, E. R. (2008). Constructive and unconstructive repetitive thought. ''Psychological Bulletin, 134''(2), 163–206. https://doi.org/10.1037/0033-2909.134.2.163

}}

==External links==
* [https://www.health.harvard.edu/mind-and-mood/break-the-cycle Breaking the cycle of rumination] (Harvard Health Publishing)
* [https://www.smilingmind.com.au/ Smiling Mind] (Smiling Mind)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Cognition]]
[[Category:Motivation and emotion/Book/Self]]</text>
      <sha1>infp2oufz70lm3x0khgwpr1918ckz0a</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Self-determination theory and military veteran reintegration</title>
    <ns>0</ns>
    <id>331144</id>
    <revision>
      <id>2834784</id>
      <parentid>2833316</parentid>
      <timestamp>2026-09-28T03:51:27Z</timestamp>
      <contributor>
        <username>U3246286</username>
        <id>3105520</id>
      </contributor>
      <comment>overview draft finished</comment>
      <origin>2834784</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="37046" sha1="6j3x1ttlbgkbotr7omo4krzpwnmty00" xml:space="preserve">{{title|Self-determination theory and military veteran reintegration:&lt;br&gt;How do autonomy, competence, and relatedness shape psychological adjustment after military service?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=5}}

;Scenario

[[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]]

What happens when the thing that gave your life structure suddenly vanishes? 

For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. 

After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. 

If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? 
{{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024).  

Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness, and relatedness by feeling connected to and valued by others (Ryan &amp; Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation.   

This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs  may shape or influence veterans' emotional wellbeing, psychological adjustment and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration.  
 

* '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''':
* SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022).
* SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022).
* This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs.
* '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments.
* Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022).
* This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions.

Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. 

* '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life.  

{{RoundBoxTop|theme=5}}

;Focus questions

{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 

2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 

3. How military experiences and reintegration shape autonomy, competence, and relatedness? 

4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration?   
{{RoundBoxBottom}}

== How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? ==

{{ic|The first two bullet points could make for an effective Overview}}

* Military-to-civilian transition involves a substantial change to veterans' occupational roles, relationships, routines, identity and their overall sense of purpose. 
* These changes challenge veterans' wellbeing, motivation, wellbeing and psychological adjustment as they adapt to a new environment. 
* Therefore understanding the psychological processes underlying this transition is essential and Self Determination Theory (SDT) provides a framework to examine how changes in the environment shape veterans' psychological needs and adjustment. 

=== What is Self Determination Theory (SDT)? ===

{{ic|Embed links to further info}}

*Self determination theory (SDT) focuses on three basic psychological needs and varied forms of motivation (autonomous to controlled). Provides a blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for psychological flourishing, positive well-being and positive quality of life (Ryan et al. 2022).
*Autonomy is the core psychological need to feel like the author of your own life, those that are autonomous want their actions to match their true values and choices rather than being forced or controlled by external pressures. Competence refers to feeling effective and satisfactory in their capacity, is not an attained skill or capability but is instead a felt sense of confidence. Relatedness refers to feeling connected to others and caring for and by others, along with having a sense of belongingness in both with other individuals and with their community (Ryan et al. 2022).
*Need satisfaction vs. need frustration. Need satisfaction drives psychological growth and well-being whilst need frustration (active blocking of these needs), can cause distress, maladaptive behaviours and ill-being (Olafsen, et al. 2021). See Table 1 below for a more in-depth explanation of these needs. 

{| class=&quot;wikitable&quot;
|+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021)
!
!Autonomy
!Competence
!Relatedness
|-
|Need satisfaction
|experiencing choice, self-direction, freedom in ones' actions
|experiencing mastery, capability and effectiveness in skills. 
|feeling a warm, secure and mutual care with others. 
|-
|Need frustration
|feeling controlled, pressured or compelled to act in an unwanted way
|feeling incompetent, facing frequent failure and doubting abilities.  
|experiencing loneliness, alienation, and social exclusion. 
|}

* SDT also focuses on different forms of motivation which range from autonomous to controlled motivation and how the outcome relates to vitality, engagement, performance and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation whilst controlled motivation involves external demands or internal pressures (Rybakovaite, et al. 2021).
* SDT offers a unique explanatory value by connecting environmental changes to motivation, wellbeing, and psychological need satisfaction. Many theories try to explain identity, resource loss or group motivation as a primary motivator, SDT helps explain why changes influence veterans' psychological adjustment. 

=== The Military Environment ===
The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. 

*Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships.
* For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing.

* Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative.

== How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? ==

* Psychology need satisfaction provides a mechanism with social environments may either positively or negatively alter their wellbeing, motivation, and psychological adjustment during their transition. 
* SDT conceptualises motivation along a continuum of self-determination, it ranges from intrinsic motivation and forms of autonomous extrinsic motivation to then controlled forms of extrinsic motivation and amotivation. 

=== Autonomy and the experience of choice ===

* SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. 
* Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. 
* Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024).
* Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024).
* Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. 
* Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. 

=== Competence and the sense of capability ===

* Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026).
*Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. 
*However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026).

=== Relatedness and the need to belong ===

* Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. 
* Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021).
* Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. 

=== Motivation and wellbeing ===

* The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026).
* Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026).
* ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' 

=== Interacting needs ===

* Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018).
* Frustration in one area may also coincide with difficulties/challenges in another. 
* ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' 

== How does military experiences and reintegration shape autonomy, competence and relatedness? ==

* The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022).
* The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. 
* Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]]
* SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. 

=== Structure, hierarchy, and autonomy ===

* Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments.
* Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021).
* Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). 
* However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021).
* This creates an important distinction between control and structure, and that leadership style and individual interpretation. 
* Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. 

&lt;quiz display=simple&gt;
{'''When can military structure support autonomy?'''

|type=&quot;()&quot;}
+ When personnel understand and value their purpose
- When all choices are removed
- When personnel experience coercion and external pressure
- When responsibilities are increased

{'''Which psychological need may be most affected when a veteran loses their military unit after discharge?'''
|type=&quot;()&quot;}
- Competence
+ Relatedness
- Belonging
- Autonomy
&lt;/quiz&gt;

=== Training, occupational roles, and competence ===

* Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. 
* Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026).
* A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026).
* This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments.
* Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention.
* From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles.
* Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment.

=== Relatedness: from military cohesion to civilian belonging ===
* Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021).
* SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging.
* Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. 
* The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021).
* A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021).
* This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one.
* ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' 

=== Identity and purpose ===

* The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. 
* Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. 
* Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined.
* Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition.
* Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings.
* Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways.  
* ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' &lt;br /&gt;

== How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? ==

* Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. 
* An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. 
* Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. 

=== Creating supportive transition environments ===

* Reintegration support should address psychological as well as practical aspects of transition
* SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. 
* Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. 
* Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. 

=== Supporting competence and connection ===

* Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. 
* Career counselling, feedback, recognition and retraining can strengthen perceived competence. 
* Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. 
* Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. 

=== Implications and limitations ===

* SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. 
* Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. 
* Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. 

;Embedded links (keeping for reference)
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

==Conclusion==
* Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. 
* SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). 
* The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. 
* Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). 
* SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. 
* Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service.   

==See also==
* [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026))
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894

Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127

Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312

Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634

Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710   

Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The &quot;Transition&quot; to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}}

Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 

Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 

Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2

Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007

Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306

Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7

Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2

Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6  

Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w

}}


APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Self-determination theory]]
[[Category:Motivation and emotion/Book/Veterans]]</text>
      <sha1>6j3x1ttlbgkbotr7omo4krzpwnmty00</sha1>
    </revision>
    <revision>
      <id>2834794</id>
      <parentid>2834784</parentid>
      <timestamp>2026-09-28T04:43:45Z</timestamp>
      <contributor>
        <username>U3246286</username>
        <id>3105520</id>
      </contributor>
      <comment>working on first focus question</comment>
      <origin>2834794</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="36780" sha1="kopw6hk9y7bg1elxnf1gmu3ce9dz73h" xml:space="preserve">{{title|Self-determination theory and military veteran reintegration:&lt;br&gt;How do autonomy, competence, and relatedness shape psychological adjustment after military service?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=5}}

;Scenario

[[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]]

What happens when the thing that gave your life structure suddenly vanishes? 

For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. 

After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. 

If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? 
{{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024).  

Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness, and relatedness by feeling connected to and valued by others (Ryan &amp; Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation.   

This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs  may shape or influence veterans' emotional wellbeing, psychological adjustment and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration.  
 

* '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''':
* SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022).
* SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022).
* This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs.
* '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments.
* Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022).
* This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions.

Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. 

* '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life.  

{{RoundBoxTop|theme=5}}

;Focus questions

{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 

2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 

3. How military experiences and reintegration shape autonomy, competence, and relatedness? 

4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration?   
{{RoundBoxBottom}}

== How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? ==

{{ic|The first two bullet points could make for an effective Overview}}

Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs, autonomy, competence, and relatedness to function effectively. 

=== What is Self Determination Theory (SDT)? ===

*
*Autonomy is the core psychological need to feel like the author of your own life, those that are autonomous want their actions to match their true values and choices rather than being forced or controlled by external pressures. Competence refers to feeling effective and satisfactory in their capacity, is not an attained skill or capability but is instead a felt sense of confidence. Relatedness refers to feeling connected to others and caring for and by others, along with having a sense of belongingness in both with other individuals and with their community (Ryan et al. 2022).
*Need satisfaction vs. need frustration. Need satisfaction drives psychological growth and well-being whilst need frustration (active blocking of these needs), can cause distress, maladaptive behaviours and ill-being (Olafsen, et al. 2021). See Table 1 below for a more in-depth explanation of these needs. 

{| class=&quot;wikitable&quot;
|+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021)
!
!Autonomy
!Competence
!Relatedness
|-
|Need satisfaction
|experiencing choice, self-direction, freedom in ones' actions
|experiencing mastery, capability and effectiveness in skills. 
|feeling a warm, secure and mutual care with others. 
|-
|Need frustration
|feeling controlled, pressured or compelled to act in an unwanted way
|feeling incompetent, facing frequent failure and doubting abilities.  
|experiencing loneliness, alienation, and social exclusion. 
|}

* SDT also focuses on different forms of motivation which range from autonomous to controlled motivation and how the outcome relates to vitality, engagement, performance and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation whilst controlled motivation involves external demands or internal pressures (Rybakovaite, et al. 2021).
* SDT offers a unique explanatory value by connecting environmental changes to motivation, wellbeing, and psychological need satisfaction. Many theories try to explain identity, resource loss or group motivation as a primary motivator, SDT helps explain why changes influence veterans' psychological adjustment. 

=== The Military Environment ===
The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. 

*Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships.
* For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing.

* Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative.

== How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? ==

* Psychology need satisfaction provides a mechanism with social environments may either positively or negatively alter their wellbeing, motivation, and psychological adjustment during their transition. 
* SDT conceptualises motivation along a continuum of self-determination, it ranges from intrinsic motivation and forms of autonomous extrinsic motivation to then controlled forms of extrinsic motivation and amotivation. 

=== Autonomy and the experience of choice ===

* SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. 
* Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. 
* Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024).
* Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024).
* Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. 
* Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. 

=== Competence and the sense of capability ===

* Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026).
*Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. 
*However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026).

=== Relatedness and the need to belong ===

* Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. 
* Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021).
* Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. 

=== Motivation and wellbeing ===

* The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026).
* Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026).
* ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' 

=== Interacting needs ===

* Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018).
* Frustration in one area may also coincide with difficulties/challenges in another. 
* ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' 

== How does military experiences and reintegration shape autonomy, competence and relatedness? ==

* The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022).
* The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. 
* Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]]
* SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. 

=== Structure, hierarchy, and autonomy ===

* Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments.
* Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021).
* Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). 
* However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021).
* This creates an important distinction between control and structure, and that leadership style and individual interpretation. 
* Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. 

&lt;quiz display=simple&gt;
{'''When can military structure support autonomy?'''

|type=&quot;()&quot;}
+ When personnel understand and value their purpose
- When all choices are removed
- When personnel experience coercion and external pressure
- When responsibilities are increased

{'''Which psychological need may be most affected when a veteran loses their military unit after discharge?'''
|type=&quot;()&quot;}
- Competence
+ Relatedness
- Belonging
- Autonomy
&lt;/quiz&gt;

=== Training, occupational roles, and competence ===

* Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. 
* Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026).
* A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026).
* This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments.
* Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention.
* From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles.
* Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment.

=== Relatedness: from military cohesion to civilian belonging ===
* Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021).
* SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging.
* Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. 
* The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021).
* A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021).
* This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one.
* ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' 

=== Identity and purpose ===

* The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. 
* Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. 
* Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined.
* Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition.
* Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings.
* Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways.  
* ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' &lt;br /&gt;

== How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? ==

* Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. 
* An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. 
* Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. 

=== Creating supportive transition environments ===

* Reintegration support should address psychological as well as practical aspects of transition
* SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. 
* Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. 
* Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. 

=== Supporting competence and connection ===

* Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. 
* Career counselling, feedback, recognition and retraining can strengthen perceived competence. 
* Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. 
* Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. 

=== Implications and limitations ===

* SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. 
* Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. 
* Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. 

;Embedded links (keeping for reference)
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

==Conclusion==
* Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. 
* SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). 
* The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. 
* Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). 
* SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. 
* Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service.   

==See also==
* [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026))
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894

Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127

Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312

Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634

Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710   

Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The &quot;Transition&quot; to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}}

Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 

Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 

Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2

Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007

Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306

Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7

Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2

Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6  

Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w

}}


APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Self-determination theory]]
[[Category:Motivation and emotion/Book/Veterans]]</text>
      <sha1>kopw6hk9y7bg1elxnf1gmu3ce9dz73h</sha1>
    </revision>
  </page>
  <page>
    <title>User:U3279062</title>
    <ns>2</ns>
    <id>331170</id>
    <revision>
      <id>2834779</id>
      <parentid>2834207</parentid>
      <timestamp>2026-09-28T02:06:11Z</timestamp>
      <contributor>
        <username>U3279062</username>
        <id>3106301</id>
      </contributor>
      <comment>added edits</comment>
      <origin>2834779</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6207" sha1="astrwmi6h76zblgdkua00fjl4b1n0gs" xml:space="preserve">
== About Me ==
Hi, I am Kayla! I am a third-year psychology student completing a double degree in Psychology and Global Studies. I am interested in psychology because I enjoy learning about the world around me and the ways psychology can connect to and inform many different areas of study. I am particularly interested in communication, learning, and abnormal psychology, although I am curious about psychology as a whole and enjoy exploring the different areas it has to offer.

As I continue through my degree, I am hoping to build on the knowledge and skills I have developed so far and become more confident in applying them. I am especially interested in developing my communication skills and learning how to connect and collaborate with others in the field. Research method and statistics have defiantly been challenging at times, my experiences with Jamovi were initially a little intimidating! but they have also encouraged me to step outside my comfort zone and continue to develop my research skills.

I am still figuring out exactly what career path I would like to pursue. Studying Psychology alongside Global Studies has given me the opportunity to explore different perspectives and possibilities, but I know that I would like psychology to be part of my future career in some way.

Outside of university, I enjoy reading, drawing and painting, and gaming. I would describe myself as curious and open-minded, and I like approaching psychology with a willingness to learn and consider different perspectives.

I am also looking forward to the social side of learning. I am friendly and open to connecting with classmates, sharing ideas, and learning from other peoples experiences. I hope this Wikiversity space can be an opportunity not only to develop academically, but also to make connections with others who are interested in psychology and learning together.

Connect with me on [https://www.linkedin.com/in/kayla-hill-316378421/ LinkedIn]

Thank you for visiting! 

== Book Chapter I'm Working On ==
Here is the link to the chapter I am working on: [[Motivation and emotion/Book/2026/Romantic jealousy|Romantic Jealousy]]

== Social Contributions and Discussions ==
Here are links to discussion posts I have created, or participated in:

# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455064 Class connect- sharing LinkedIn profiles]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455468 Space for topic development questions]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345?entry_id=806021 'How do you motivate yourself?' answer]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261?entry_id=804351 'What do you really want to learn about?' answer]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456676 Book chapter hub]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Use of AI question] 

Here are links to direct wiki edits:

# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Attachment_styles_and_relatedness_motivation&amp;diff=prev&amp;oldid=2824585 Attachment style and relatedness edit]  
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAthletic_identity_loss_and_returning_to_sport_after_injury&amp;diff=2832195&amp;oldid=2828842 Athletic injury edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAdolescent_risk-taking_and_reward-system_development&amp;diff=2834188&amp;oldid=2829686 Adolescent risk-taking edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAkrasia&amp;diff=2834189&amp;oldid=2833141 Akrasia edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAutomaticity_and_goal_pursuit&amp;diff=2834193&amp;oldid=2831941 Automaticity and goal pursuit edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FCompetence_motivation_in_self-determination_theory&amp;diff=2834194&amp;oldid=2833291 Competence motivation in self-determination theory edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FConsumer_emotion_measurement&amp;diff=2834195&amp;oldid=2831255 Consumer emotion measurement edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FDeveloping_a_growth_mindset&amp;diff=2834196&amp;oldid=2833184 Developing a growth mindset edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEffort_regulation_and_cost-benefit_decision-making&amp;diff=2834197&amp;oldid=2833107 Effort regulation and cost-benefit decision-making edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/ERG_theory_and_motivation&amp;action=history ERG theory and motivation edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExercise_gamification_motivation&amp;diff=2834200&amp;oldid=2834140 Exercise gamification motivation edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExpectancy-value_theory_of_educational_motivation&amp;diff=2834203&amp;oldid=2831722 Expectancy-value theory of educational motivation edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExtended_process_model_of_emotion_regulation&amp;diff=2834205&amp;oldid=2833718 Extended process model of emotion regulation edit] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FFuture_orientation_and_criminal_behaviour&amp;diff=2834772&amp;oldid=2833305 Future orientation and criminal behaviour] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FImpulsivity_versus_sensation-seeking&amp;diff=2834774&amp;oldid=2834696 Impulsivity versus sensation-seeking] 
# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FMindsets_and_stigma&amp;diff=2834776&amp;oldid=2834456 Mindsets and stigma] 

Here are links to wiki discussions: 

# [[Talk:Motivation and emotion/Book/2026/Parental motivations for homeschooling#c-U3279062-20260823083200|Parental motivations for homeschooling discussion]]</text>
      <sha1>astrwmi6h76zblgdkua00fjl4b1n0gs</sha1>
    </revision>
  </page>
  <page>
    <title>User:StretchBeyond</title>
    <ns>2</ns>
    <id>331218</id>
    <revision>
      <id>2834788</id>
      <parentid>2834632</parentid>
      <timestamp>2026-09-28T04:07:49Z</timestamp>
      <contributor>
        <username>StretchBeyond</username>
        <id>3105744</id>
      </contributor>
      <comment>/* Social Contributions */</comment>
      <origin>2834788</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11373" sha1="1qp4sxksi80qd8fstis2m1yo0wutaia" xml:space="preserve">== About Me ==
I am a military veteran, working to complete a Bachelor of Science in Psychology as part of a lifelong approach to learning. After a military career working with people around the world on a spectrum of operations, I wanted to better understand the human dimension and the science behind what makes us all tick. Along the way, I want to give back to the community where I can. I have an interest in veterans' PTSD, advocacy and international affairs.

== Book Chapter ==
[[File:PTSD.png|thumb|250x250px|'''Figure 1.''' PTSD can have a deep and lasting impact on our emotions.]]
I am writing a book chapter titled [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|Immersive Therapy for PTSD: How does it work and what are the effects?]] 

This chapter is an individual student contribution as part of a larger [https://www.canberra.edu.au/ University of Canberra] class effort in 2026 to write a book [[Motivation and emotion/Book]]. 

The 2026 book extends a Wikiversity book series started in 2010, with over 1,800 online book chapters on how psychological science can improve human lives.

The book editor and University psychology unit convener is [[User:Jtneill|Dr. James Neill]]

==== Chapters I am working on: ====

====== [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|Immersive Therapy for PTSD: How does it work and what are the effects?]] ======

====== [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment/Non-invasive brain stimulation techniques|High-frequency rTMS and 3MDR in the treatment of PTSD: An integrated physiological and clinical review]] ======
{{clear}}

== Social Contributions ==

#'''18 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Defence mechanisms and emotion regulation. Edits at https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FDefence_mechanisms_and_emotion_regulation&amp;diff=2822629&amp;oldid=2763661
# '''18 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Neurodivergence and trauma. Edits at https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FNeurodivergence_and_trauma&amp;diff=2822636&amp;oldid=2812805
# '''19 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Incentive theory of motivation. Edits at&lt;nowiki/&gt;https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FIncentive_theory_of_motivation&amp;diff=2822844&amp;oldid=2814898
# '''19 Aug 26:''' UC Canvas Discussion: Introduced the book chapter intent to the class and welcomed anyone with sim&lt;nowiki/&gt;ilar topics or interests to contribute. I offered the same. https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456450
# '''20 Aug 26:''' Comment: Supported another Wiki user by providing some peer-reviewed ideas on starter references f&lt;nowiki/&gt;or a Chapter page. [[Talk:Motivation and emotion/Book/2026/Perfectionism and procrastination#Reference material]]
# '''20 Aug 26:''' Comment: Supported another Wiki user by providing some peer-reviewed ideas on starter references f&lt;nowiki/&gt;or a Chapter page.[[Talk:Motivation and emotion/Book/2026/Irritability#Reference material]]
# '''22 Aug 26:''' Grammar and content editing improvements on 2025 chapter, Mental health in astronauts. Edits at [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMental_health_in_astronauts&amp;diff=2824159&amp;oldid=2804667 ht]&lt;nowiki/&gt;[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMental_health_in_astronauts&amp;diff=2824159&amp;oldid=2804667 tps://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMental_health_in_astronauts&amp;diff=2824159&amp;oldid=2804667]
# '''25 Aug 26:''' Comment: Supported another Wiki user by providing some peer-reviewed ideas on starter references f&lt;nowiki/&gt;or a Chapter page.[[Talk:Motivation and emotion/Book/2026/Feedback literacy#Reference]]
# '''25 Aug 26''': UC Canvas Discussion: Responded to student topic query.[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 https://uclearn.canberra.edu.au/courses/201]&lt;nowiki/&gt;[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 43/discussion_topics/457190]
# '''26 Aug 26:''' Comment: Supported another Wiki user by providing some external link ideas for a Chapter page.[[Talk:Motivation and emotion/Book/2026/Dark empathy#Reference|Talk]]&lt;nowiki/&gt;[[Talk:Motivation and emotion/Book/2026/Dark empathy#Reference|:Motivation and emotion/Book/2026/Dark empathy#Reference]]
# '''26 Aug 26:''' UC Canvas Discussion: Responded to student topic query with data from a Random Game of Dice scenar&lt;nowiki/&gt;io.https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457412
# '''27 Aug 26:''' Wiki thanks and editing conversation: [[User talk:Jtneill#Punctuation change in book title]]
# '''28 Aug 26:''' UC Canvas Discussion: Responded to student topic query&lt;nowiki/&gt; https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457809
# '''4 Sep 26:''' Comment: Included creation of graphics for Wiki page.[[c:File:Avoidance_and_Processing_Cycles.png|ht]]&lt;nowiki/&gt;[[c:File:Avoidance_and_Processing_Cycles.png|tps://commons.wikimedia.org/wiki/File:Avoidance_and_Processing_Cycles.png]]
# '''4 Sep 26:''' Comment: Assisted other Wiki user by adding a quiz temp&lt;nowiki/&gt;late to their chapter page https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FBuilding_therapeutic_alliance&amp;diff=2831108&amp;oldid=2830050
# '''8 Sep 26:''' Comment: Assisted other Wiki user by adding a quiz temp&lt;nowiki/&gt;late to their chapter page https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-concept_and_motivation&amp;diff=2832141&amp;oldid=2831622
# '''9 Sep 26:''' Comment: Assisted another Wiki user by adding a quiz te&lt;nowiki/&gt;mplate to their chapter page&lt;nowiki/&gt;https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FOutdoor_play_and_children%27s_emotional_well-being&amp;diff=2832147&amp;oldid=2828722
# '''9 Sep 26:''' UC Canvas Discussion: Responded to student topic query &lt;nowiki/&gt;https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457888
# '''11 Sep 26:''' UC Canvas Discussion: Chapter word count question - in&lt;nowiki/&gt;cludes great response advice from other contributors on tools and techniques https://uclearn.canberra.edu.au/courses/20143/discussion_topics/461216
# '''14 Sep:''' Comment: Assisted another Wiki user by adding some advice&lt;nowiki/&gt; on prediction error references [[Talk:Motivation and emotion/Book/2026/Fear extinction#Prediction Error and Extinction]]
# '''15 Sep 26:''' Correspondence: Email exchange between chapter author &lt;nowiki/&gt;and Professor Vermetten, who is credited with creating the 3MDR therapy and referenced in the chapter.  &quot;I have looked at the Wikiversity page and think you are on the right path. You have brought together the concepts of immersion, embodiment, motivation, and trauma-focused treatment in a thoughtful and relevant way.&quot;
# '''16 Sep 26:''' Grammar and content editing improvements on 2025 chapt&lt;nowiki/&gt;er due to links with PTSD and prediction error ,https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FNeuroscience_of_unexpected_positive_outcomes&amp;diff=2833368&amp;oldid=2762078
# '''16 Sep 26:''' UC Canvas Discussion: Responded to student request for&lt;nowiki/&gt; chapter feedback on impulsivity and sensation seeking https://uclearn.canberra.edu.au/courses/20143/discussion_topics/461607
# '''16 Sep 26:''' UC Canvas Discussion: Responded to student request for&lt;nowiki/&gt; chapter feedback on brain implants for chronic pain&lt;nowiki/&gt;https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457902
# '''17 Sep 26:''' UC Canvas Discussion: Chapter referencing question abo&lt;nowiki/&gt;ut integration of non-academic journal sources neede&lt;nowiki/&gt;d to source financial market analysis relating to immersive PTSD forecasts. https://uclearn.canberra.edu.au/courses/20143/discussion_topics/462166
# '''21 Sep 26:''' Comment: Assisted another Wiki user by making formatti&lt;nowiki/&gt;ng changes in chapter heading styles to create page &lt;nowiki/&gt;menu. https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FHypothalamus_and_homeostatic_motivation&amp;diff=2834144&amp;oldid=2833547
# '''21 Sep 26:''' UC Canvas discussion: Chapter feedback provided on UC &lt;nowiki/&gt;Canvas at the request of the author.[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457901 https://uclearn.]&lt;nowiki/&gt;[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457901 canberra.edu.au/courses/20143/discussion_topics/457901]
# '''21 Sep 26:''' UC Canvas discussion: Assisted user with search for li&lt;nowiki/&gt;terature. Provided a number of contemporary meta-ana&lt;nowiki/&gt;lyses and systetemic reviews from the library that may help their research. https://uclearn.canberra.edu.au/courses/20143/discussion_topics/459237
# '''23 Sep 26:''' UC Canvas discussion: From combat to cognition - a lay&lt;nowiki/&gt;person's insights into motivation [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345 https://uclearn.ca]&lt;nowiki/&gt;[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345 nberra.edu.au/courses/20143/discussion_topics/456345]
# '''24 Sep 26:''' UC Canvas discussion: What do you really want to learn&lt;nowiki/&gt; about - Body language and non-verbal communication[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 h]&lt;nowiki/&gt;[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 ttps://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261]
# '''25 Sep 26:''' Added sub-page content to consolidate additional readi&lt;nowiki/&gt;ng, learning and research that I have done between P&lt;nowiki/&gt;TSD treatment and physiological psychology involving non-invasive brain stimulation: [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment/Non-invasive brain stimulation techniques]]
# '''27 Sep 26:''' Wiki editing of another chapter: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-concept_and_motivation&amp;diff=2834628&amp;oldid=2834606 https://en.wikiversit]&lt;nowiki/&gt;[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-concept_and_motivation&amp;diff=2834628&amp;oldid=2834606 y.org/w/index.php?title=Motivation_and_emotion%2FBoo]&lt;nowiki/&gt;[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-concept_and_motivation&amp;diff=2834628&amp;oldid=2834606 k%2F2026%2FSelf-concept_and_motivation&amp;diff=2834628&amp;oldid=2834606]
# '''27 Sep 26:''' UC Canvas discussion: Chapter feedback provided on UC &lt;nowiki/&gt;Canvas at the request of the author [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/463438 https://uclearn.]&lt;nowiki/&gt;[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/463438 canberra.edu.au/courses/20143/discussion_topics/463438]
# '''28 Sep 26:''' Wiki discussion page - thanked another user for their &lt;nowiki/&gt;excellent, constructive feedback.[[Talk:Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|Talk:Motivation and]]&lt;nowiki/&gt; [[Talk:Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|emotion/Book/2026/Immersive therapy for PTSD treatment]]</text>
      <sha1>1qp4sxksi80qd8fstis2m1yo0wutaia</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being</title>
    <ns>0</ns>
    <id>331234</id>
    <revision>
      <id>2834673</id>
      <parentid>2834669</parentid>
      <timestamp>2026-09-27T12:31:17Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Working memory and cognitive process */</comment>
      <origin>2834673</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="46808" sha1="kicky22v7xex5eupidteoarujzkjv9e" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|100x100px|thumb]]

'''Focus questions'''

* What are outdoor play and children's emotional well-being?
* How does outdoor play influence children's emotional well-being?
* How do social and environmental conditions shape  children's outdoor play experiences?
* How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action provided by features of an environment (Morgenthaler et al., 2024). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides to individuals. The theory provides a way of understanding on how features of outdoor environment can provide opportunities for children to interact with different forms of play . Morgenthaler et al. (2024) applied the theory of affordances to examine environment play transactions and development of an environmental taxonomy of outdoor play space features. The review identified 284 different play opportunities afforded by different spaces and objects. This suggests that outdoor environment not only provides a space for play, but the environment's physical features can provide different opportunities for how children interact with and use the play space.

Research on outdoor play space supports the importance of these environmental features. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review included features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. The study found that physical activity was the most commonly examined outcome. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features. Therefore, these findings show that outdoor play differ in the opportunities they provide, but the evidence does not support that one particular features produces better emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, social play, active play
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play [''add a reference]''. Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  Affective social competence provides a psychological framework for understanding these social interactions. Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play.

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, it is noteworthy that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being.

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play [''add a reference]''. Beekhuizen et al. (2025) identified [''identified what?]'' as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with, children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during game play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when child needed help to access or participate in outdoor play. Time constraints also affect opportunities for outdoor play. Some children have less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school. These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation. Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important to children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024)  found that the evidence for individual outdoor play space features was inconsistent across studies. Differences in study methods and outcomes made it difficult to determine whether particular features were linked to children's behaviour. Gibson et al. (2017) reported a similar limitation for loose-parts play, with insufficient quantitative evidence to determine whether these interventions improved children's cognitive, social or emotional development. Physical access also does not mean that all children experience the same opportunities for play. Beekhuizen et al. (2025) found that accessibility was particularly relevant for children with disabilities because features of the play environment could support or restrict participation. Together, these findings show that environmental features can shape opportunities for play, but current evidence does not establish that particular feature improves children's emotional well-being. 

{{ic|Add an APA style table caption}}

{| class=&quot;wikitable&quot;
|+
|Key points   

* Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could facilitate or restrict inclusive outdoor play.
* Social relationships and familiarity affected participation, with children with disabilities reporting more barriers to inclusive outdoor play (Beekhuizen et al., 2025).
* Associations between outdoor play and social-emotional competence varied across some population groups and play contexts (Ferguson et al., 2025).
|}

== '''What does recent research show about outdoor play and children's emotional well-being?''' ==

* Recent research has examined various aspects of outdoor play and children's emotional well-being and provides mixed evidence. These include emotional dysregulation (Lee et al., 2025), social-emotional competence (Ferguson et al., 2025), positive and negative affect (Poulos et al., 2026), and internalising and externalising symptoms (Dodd et al., 2026).
* Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. However, the associations varied across some ethnic groups and according to whether outdoor play occurred on school days or weekends.
* Dodd et al. (2026) provides longitudinal evidence from 4,151 children from the Growing Up in Scotland study. Children who played outdoors more frequently between approximately 2 and 4 years of age were more likely to follow a trajectory of low and stable internalising and externalising symptoms through to age 8.
* Poulos et al. (2026) examined children's emotional state immediately after recess during hot weather. Among 317 children in years 4 and 5, the positive effect was higher after outdoor recess and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings.
* Together, these studies show evidence of associations between outdoor play or outdoor recess and several aspects of emotional well-being. However, the studies did not show the common effect because the study differ in children's ages, research designs, measures of outdoor play, and emotional outcomes.

=== '''Consistency of recent research findings''' ===
* Recent findings do not show the same effect across all outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with lower total difficulties and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. Therefore, the recent studies have not found significant association across all developmental outcomes.
* Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that lower emotional dysregulation is associated with greater afternoon outdoor play but higher emotional dysregulation is associated with greater outdoor play earlier and later in the day.
* Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years predicted lower odds of belonging to the increasing or decreasing internalising and externalising symptom trajectory groups, relative to the normative low and stable group. This shows an association between early outdoor play and mental-health symptom trajectories across childhood rather than an emotional outcome measured at one point in time.
* Associations may also vary across children and outdoor play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends.
* Recent findings suggest that associations between outdoor play and emotional well-being are not the same across all outcomes, children and play contexts. The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===

* Longitudinal research strengthens the evidence concerning temporal relationship. Dodd et al. (2026) analysed 4,151 children from the Growing Up in Scotland study. Outdoor play was measured at approximately 2, 3 and 4 years of age, while internalising and externalising symptoms were measured at approximately 4, 5, 6 and 8 years. This allowed the researchers to examine whether early outdoor play predicted different mental-health symptom trajectories across childhood.
* Although, longitudinal evidence does not establish cause and effect. Dodd et al. (2026) identified reliance on parent report measures as a key limitation, while Lee et al. (2025) also discussed about the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation.
* Measurement of outdoor play is another important consideration. Davenport et al. (2024) examined both parent reported and device measured outdoor play and also tested whether associations were independent of outdoor moderate to vigorous physical activity. This approach distinguishes outdoor play from outdoor physical activity rather than interchangeable measures.
* Limitations are also present across the wider outdoor play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor play publications and found that cross sectional study design was most common. Mental and emotional development had received less research attention in comparison to physical health and development.

{| class=&quot;wikitable&quot;
|+Table 2. Comparison of recent research
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; the authors also considered possible reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample , and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Outdoor play and symptoms relied on parent reported measures, and the observational design does not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared children's positive and negative affect following outdoor, gymnasium and classroom recess rather than using outdoor recess without a comparison setting.
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 2, the recent evidence has different methodological strengths and limitations. Longitudinal data provide stronger evidence about temporal ordering than cross-sectional studies, while the use of parent-reported and device-measured outdoor play addresses some measurement concerns. However, much of the evidence remains observational, several studies rely on parent-reported measures, and differences in emotional outcomes and play contexts limit  the direct comparison between studies.  

{| class=&quot;wikitable&quot;
|Key points

* Recent studies report positive, mixed and non-significant findings across different emotional outcomes and measures of outdoor play (Davenport et al., 2024; Ferguson et al., 2025).
* Lee et al. (2025) and Dodd et al. (2026) found associations across different emotional outcomes and time frames, from emotional dysregulation to later symptom trajectories.
* Differences in study design, measurement, children's age and play context mean the findings should not be interpreted as one consistent effect of outdoor play on emotional well-being. 

|}

== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>kicky22v7xex5eupidteoarujzkjv9e</sha1>
    </revision>
    <revision>
      <id>2834737</id>
      <parentid>2834673</parentid>
      <timestamp>2026-09-27T23:39:34Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* How does social and environmental conditions influence on children's outdoor play experiences? */</comment>
      <origin>2834737</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47944" sha1="e1496sa8r77jdpg7hpiiy6rrlcdws8u" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|100x100px|thumb]]

'''Focus questions'''

* What are outdoor play and children's emotional well-being?
* How does outdoor play influence children's emotional well-being?
* How do social and environmental conditions shape  children's outdoor play experiences?
* How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==

* Recent research has examined various aspects of outdoor play and children's emotional well-being and provides mixed evidence. These include emotional dysregulation (Lee et al., 2025), social-emotional competence (Ferguson et al., 2025), positive and negative affect (Poulos et al., 2026), and internalising and externalising symptoms (Dodd et al., 2026).
* Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. However, the associations varied across some ethnic groups and according to whether outdoor play occurred on school days or weekends.
* Dodd et al. (2026) provides longitudinal evidence from 4,151 children from the Growing Up in Scotland study. Children who played outdoors more frequently between approximately 2 and 4 years of age were more likely to follow a trajectory of low and stable internalising and externalising symptoms through to age 8.
* Poulos et al. (2026) examined children's emotional state immediately after recess during hot weather. Among 317 children in years 4 and 5, the positive effect was higher after outdoor recess and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings.
* Together, these studies show evidence of associations between outdoor play or outdoor recess and several aspects of emotional well-being. However, the studies did not show the common effect because the study differ in children's ages, research designs, measures of outdoor play, and emotional outcomes.

=== '''Consistency of recent research findings''' ===
* Recent findings do not show the same effect across all outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with lower total difficulties and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. Therefore, the recent studies have not found significant association across all developmental outcomes.
* Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that lower emotional dysregulation is associated with greater afternoon outdoor play but higher emotional dysregulation is associated with greater outdoor play earlier and later in the day.
* Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years predicted lower odds of belonging to the increasing or decreasing internalising and externalising symptom trajectory groups, relative to the normative low and stable group. This shows an association between early outdoor play and mental-health symptom trajectories across childhood rather than an emotional outcome measured at one point in time.
* Associations may also vary across children and outdoor play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends.
* Recent findings suggest that associations between outdoor play and emotional well-being are not the same across all outcomes, children and play contexts. The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===

* Longitudinal research strengthens the evidence concerning temporal relationship. Dodd et al. (2026) analysed 4,151 children from the Growing Up in Scotland study. Outdoor play was measured at approximately 2, 3 and 4 years of age, while internalising and externalising symptoms were measured at approximately 4, 5, 6 and 8 years. This allowed the researchers to examine whether early outdoor play predicted different mental-health symptom trajectories across childhood.
* Although, longitudinal evidence does not establish cause and effect. Dodd et al. (2026) identified reliance on parent report measures as a key limitation, while Lee et al. (2025) also discussed about the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation.
* Measurement of outdoor play is another important consideration. Davenport et al. (2024) examined both parent reported and device measured outdoor play and also tested whether associations were independent of outdoor moderate to vigorous physical activity. This approach distinguishes outdoor play from outdoor physical activity rather than interchangeable measures.
* Limitations are also present across the wider outdoor play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor play publications and found that cross sectional study design was most common. Mental and emotional development had received less research attention in comparison to physical health and development.

{| class=&quot;wikitable&quot;
|+Table 2. Comparison of recent research
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; the authors also considered possible reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample , and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Outdoor play and symptoms relied on parent reported measures, and the observational design does not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared children's positive and negative affect following outdoor, gymnasium and classroom recess rather than using outdoor recess without a comparison setting.
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 2, the recent evidence has different methodological strengths and limitations. Longitudinal data provide stronger evidence about temporal ordering than cross-sectional studies, while the use of parent-reported and device-measured outdoor play addresses some measurement concerns. However, much of the evidence remains observational, several studies rely on parent-reported measures, and differences in emotional outcomes and play contexts limit  the direct comparison between studies.  

{| class=&quot;wikitable&quot;
|Key points

* Recent studies report positive, mixed and non-significant findings across different emotional outcomes and measures of outdoor play (Davenport et al., 2024; Ferguson et al., 2025).
* Lee et al. (2025) and Dodd et al. (2026) found associations across different emotional outcomes and time frames, from emotional dysregulation to later symptom trajectories.
* Differences in study design, measurement, children's age and play context mean the findings should not be interpreted as one consistent effect of outdoor play on emotional well-being. 

|}

== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>e1496sa8r77jdpg7hpiiy6rrlcdws8u</sha1>
    </revision>
    <revision>
      <id>2834756</id>
      <parentid>2834737</parentid>
      <timestamp>2026-09-28T01:15:28Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* What does recent research show about outdoor play and children's emotional well-being? */</comment>
      <origin>2834756</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48405" sha1="flgxaftyigug626fepwfqmkeggw43jp" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|100x100px|thumb]]

'''Focus questions'''

* What are outdoor play and children's emotional well-being?
* How does outdoor play influence children's emotional well-being?
* How do social and environmental conditions shape  children's outdoor play experiences?
* How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  


Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

{| class=&quot;wikitable&quot;
|+Table 3. Comparison of recent research                                                                                                                                                 
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies.  

{| class=&quot;wikitable&quot;
|Key points

* Recent studies report positive, mixed and non-significant findings across different emotional outcomes and measures of outdoor play (Davenport et al., 2024; Ferguson et al., 2025).
* Lee et al. (2025) and Dodd et al. (2026) found associations across different emotional outcomes and time frames, from emotional dysregulation to later symptom trajectories.
* Differences in study design, measurement, children's age and play context mean the findings should not be interpreted as one consistent effect of outdoor play on emotional well-being. 

|}

== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>flgxaftyigug626fepwfqmkeggw43jp</sha1>
    </revision>
    <revision>
      <id>2834758</id>
      <parentid>2834756</parentid>
      <timestamp>2026-09-28T01:20:20Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Strengths and limitations of current research */</comment>
      <origin>2834758</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48390" sha1="f2zsi51rh874i57e0jslkexdupenhpo" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. Comparison of recent research'''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies.  

{| class=&quot;wikitable&quot;
|Key points

* Recent studies report positive, mixed and non-significant findings across different emotional outcomes and measures of outdoor play (Davenport et al., 2024; Ferguson et al., 2025).
* Lee et al. (2025) and Dodd et al. (2026) found associations across different emotional outcomes and time frames, from emotional dysregulation to later symptom trajectories.
* Differences in study design, measurement, children's age and play context mean the findings should not be interpreted as one consistent effect of outdoor play on emotional well-being. 

|}

== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>f2zsi51rh874i57e0jslkexdupenhpo</sha1>
    </revision>
    <revision>
      <id>2834759</id>
      <parentid>2834758</parentid>
      <timestamp>2026-09-28T01:21:16Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Understanding outdoor play and children's emotional well-being */</comment>
      <origin>2834759</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48392" sha1="egawgzw2n026mdthr8fktsa06v3uxc8" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. Comparison of recent research'''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies.  

{| class=&quot;wikitable&quot;
|Key points

* Recent studies report positive, mixed and non-significant findings across different emotional outcomes and measures of outdoor play (Davenport et al., 2024; Ferguson et al., 2025).
* Lee et al. (2025) and Dodd et al. (2026) found associations across different emotional outcomes and time frames, from emotional dysregulation to later symptom trajectories.
* Differences in study design, measurement, children's age and play context mean the findings should not be interpreted as one consistent effect of outdoor play on emotional well-being. 

|}

== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>egawgzw2n026mdthr8fktsa06v3uxc8</sha1>
    </revision>
    <revision>
      <id>2834760</id>
      <parentid>2834759</parentid>
      <timestamp>2026-09-28T01:21:53Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Understanding outdoor play and children's emotional well-being */</comment>
      <origin>2834760</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48414" sha1="jbxzbs9f0vtrucqb1rmv16h7vo45svf" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. Comparison of recent research'''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies.  

{| class=&quot;wikitable&quot;
|Key points

* Recent studies report positive, mixed and non-significant findings across different emotional outcomes and measures of outdoor play (Davenport et al., 2024; Ferguson et al., 2025).
* Lee et al. (2025) and Dodd et al. (2026) found associations across different emotional outcomes and time frames, from emotional dysregulation to later symptom trajectories.
* Differences in study design, measurement, children's age and play context mean the findings should not be interpreted as one consistent effect of outdoor play on emotional well-being. 

|}

== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>jbxzbs9f0vtrucqb1rmv16h7vo45svf</sha1>
    </revision>
    <revision>
      <id>2834761</id>
      <parentid>2834760</parentid>
      <timestamp>2026-09-28T01:23:24Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Strengths and limitations of current research */</comment>
      <origin>2834761</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48418" sha1="4eunrop3tnvcvnkxpyai6ix4khbu5kh" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies.  

{| class=&quot;wikitable&quot;
|Key points

* Recent studies report positive, mixed and non-significant findings across different emotional outcomes and measures of outdoor play (Davenport et al., 2024; Ferguson et al., 2025).
* Lee et al. (2025) and Dodd et al. (2026) found associations across different emotional outcomes and time frames, from emotional dysregulation to later symptom trajectories.
* Differences in study design, measurement, children's age and play context mean the findings should not be interpreted as one consistent effect of outdoor play on emotional well-being. 

|}

== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>4eunrop3tnvcvnkxpyai6ix4khbu5kh</sha1>
    </revision>
    <revision>
      <id>2834767</id>
      <parentid>2834761</parentid>
      <timestamp>2026-09-28T01:37:14Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Schools */</comment>
      <origin>2834767</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48420" sha1="192dw1nela7ymyc730vheqb338pon8o" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>192dw1nela7ymyc730vheqb338pon8o</sha1>
    </revision>
    <revision>
      <id>2834773</id>
      <parentid>2834767</parentid>
      <timestamp>2026-09-28T01:45:49Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Strengths and limitations of current research */</comment>
      <origin>2834773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48635" sha1="4s1iiokhafrbzv5osdqgf0zuozm2v5w" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 

Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==

* Opportunities for outdoor play are influenced by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors that could act as facilitators or barriers to inclusive outdoor play.
* Support for outdoor play also involves whether children can participate with other children. Beekhuizen et al. (2025) identified experiencing autonomy,  growing up together, making contact and adapting ways of playing as factors involved in inclusive outdoor play.
* The  2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play in settings where people live, learn, work and play. It also emphasises collaboration across sectors, settings and societies to support equitable access to active outdoor play (Lee et al., 2025).

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>4s1iiokhafrbzv5osdqgf0zuozm2v5w</sha1>
    </revision>
    <revision>
      <id>2834780</id>
      <parentid>2834773</parentid>
      <timestamp>2026-09-28T02:15:48Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* How do families, communities and schools support outdoor play opportunities? */</comment>
      <origin>2834780</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48484" sha1="raf80kojoteq2iuzrup46kjc74orsat" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 

Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==
Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation.

The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play. It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools.

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>raf80kojoteq2iuzrup46kjc74orsat</sha1>
    </revision>
    <revision>
      <id>2834781</id>
      <parentid>2834780</parentid>
      <timestamp>2026-09-28T02:16:20Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* How do families, communities and schools support outdoor play opportunities? */</comment>
      <origin>2834781</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48483" sha1="drzb4h0zyhlo0idtd7zkblpq8931yoa" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 

Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==
Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play. It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools.

=== '''Families''' ===

* Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation and, for some children, helping to explain their disability to other children.
* Experiencing autonomy was identified as an important personal factor. Some children with disabilities wanted their parents to allow them freedom during outdoor play. Some preferred their parents not to accompany them to the playground, while others were comfortable with parents remaining at a reasonable distance (Beekhuizen et al., 2025).
* These findings indicate that parental support and children's autonomy can both be relevant to opportunities for outdoor play. For some children with disabilities, support could be necessary for participation while opportunities to play more independently were also valued (Beekhuizen et al., 2025).
* Beekhuizen et al. (2025) identified parental support as important for facilitating positive inclusive play experiences. The authors also recommended providing children and parents with knowledge about disabilities and possible adaptations that can support participation
* Family level opportunities also need to be interpreted alongside the time constraints. Some children attending special education described long journeys between school and home that reduced the time available for outdoor play (Beekhuizen et al., 2025). This means that opportunities for outdoor play cannot be interpreted only in terms of children's or parents' preferences.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>drzb4h0zyhlo0idtd7zkblpq8931yoa</sha1>
    </revision>
    <revision>
      <id>2834818</id>
      <parentid>2834781</parentid>
      <timestamp>2026-09-28T07:20:20Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Families */</comment>
      <origin>2834818</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49327" sha1="rhwd9px570gzkcnandaff8iygrvfp0z" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 

Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==
Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play. It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools.

=== '''Families''' ===
Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).  

Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors  (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. 

Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer. Han et al. (2022) identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. 

Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive.

=== '''Communities''' ===

* Beekhuizen et al. (2025) identified communities and the outdoor play environment as environmental factors in inclusive outdoor play. Children described physically and socially safe playgrounds as important. While barriers included structure of playgrounds that made participation difficult for some children with disabilities.
* The social environment of a community was also important. Children with disabilities described smaller communities, such as villages and schools, the places where other children and parents could become more familiar with disability. This familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025).
* Recent observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor play sessions and assessed both children's playfulness and environmental supportiveness. Children with disabilities showed significantly lower playfulness scores indicating that opportunities for inclusive play were not experienced equally.
* The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors, settings and societies to preserve, promote and value equitable access to active outdoor play. This places responsibility for outdoor play opportunities across communities and wider systems rather than on children and families alone (Lee et al., 2025).

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>rhwd9px570gzkcnandaff8iygrvfp0z</sha1>
    </revision>
    <revision>
      <id>2834819</id>
      <parentid>2834818</parentid>
      <timestamp>2026-09-28T07:46:55Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Communities */</comment>
      <origin>2834819</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49968" sha1="mofadk217sjipm11a8489dy405li2aa" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater prosocial behaviour. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, cognitive and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 

Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==
Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play. It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools.

=== '''Families''' ===
Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).  

Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors  (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. 

Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. 

Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive.

=== '''Communities''' ===
Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser &amp; van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings.

The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate.

The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being.

=== '''Schools''' ===

* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>mofadk217sjipm11a8489dy405li2aa</sha1>
    </revision>
    <revision>
      <id>2834843</id>
      <parentid>2834819</parentid>
      <timestamp>2026-09-28T11:55:04Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <origin>2834843</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="52122" sha1="cgl59xe47fhrihzs3lmim0wd2qv2lnh" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 

Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==
[[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]]
Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools.

=== '''Families''' ===
Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).  

Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors  (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. 

Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. 

Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive.

=== '''Communities''' ===
Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser &amp; van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings.

The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate.

The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being.

=== '''Schools''' ===
Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints.

The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. 

Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. 
* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|1. Children who have access to the same outdoor playground may still have different opportunities to participate in play.

True                                                                                                                                                                                
False                                                                                                                                                                                                 
2. Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.                                                                                                                                                               
True                                                                                                                                                                                
False                                                                                                                                                                              

|}

{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}
'''''&lt;u&gt;Note: Answers to the quiz is 1: True and 2 False&lt;/u&gt;'''''

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>cgl59xe47fhrihzs3lmim0wd2qv2lnh</sha1>
    </revision>
    <revision>
      <id>2834844</id>
      <parentid>2834843</parentid>
      <timestamp>2026-09-28T11:57:13Z</timestamp>
      <contributor>
        <username>Mymunu</username>
        <id>3106327</id>
      </contributor>
      <comment>/* Conclusion */</comment>
      <origin>2834844</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="50862" sha1="j8w1f8r8e3eg9ucf2j0m3olklfidu8y" xml:space="preserve">{{title|Outdoor play and children's emotional well-being:&lt;br&gt;How does outdoor play influence children's emotional well-being?}}
__TOC__

== Overview ==
{| class=&quot;wikitable&quot;
[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]]

|'''Case study: Part 1'''

A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015).
|}

Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional  situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. 

{| class=&quot;wikitable&quot;
|

;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]]


'''Focus questions'''

* '''What are outdoor play and children's emotional well-being?'''
* '''How does outdoor play influence children's emotional well-being?'''
* '''How do social and environmental conditions shape  children's outdoor play experiences?'''
* '''What does recent research show about outdoor play and children's emotional well-being?'''
* '''How can families, communities and school support outdoor  play opportunities that may benefit children's emotional well-being?'''
|}

== '''Understanding outdoor play and children's emotional well-being''' ==
[[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]]

Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. 

=== Outdoor play ===
Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' 

=== Children's emotional well-being ===
Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared.

'''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being''
{| class=&quot;wikitable&quot;
|+
!Study
!Outcome examined
!What the outcome represents                                                                                                  
|-
|Lee et al. (2025)   
|Emotional dysregulation
|Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play
|-
|Ferguson et al.          (2025)
|Social-emotional competence 
|Social and emotional difficulties and prosocial behaviour
|-
|Poulos et al. (2026)
|Positive and negative affect
|Children's affect following recess in different settings 
|-
|Dodd et al. (2026)
|Internalising and externalising symptoms
|Internalising: emotional symptoms and peer problems
Externalising: conduct problems and hyperactivity- inattention 
|}
As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. 

'''Case study: part 2'''
 
''What is being measured?''

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}

|}

== '''How does outdoor play influence children's emotional well-being?''' ==
Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being.

Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being.

=== '''Emotion regulation''' ===
Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation.

Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. 

{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play.

{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?.
|type=&quot;()&quot;}
-A. Outdoor play causes better emotion regulation.
+B. Situation modification can help explain how the child responds to the situation.
-C. Children regulate emotions better outdoors than indoors.
-D. Stepping away from play always improves emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Working memory and cognitive process''' ===
Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation.

When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation.

Johnstone et al (2022) reviewed 36 studies of nature-based early childhood education. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including self-regulation, while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's emotional well-being.

== '''How does social and environmental conditions influence on children's outdoor play experiences?''' ==
Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through affordance theory, which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). 

=== Affordance theory and outdoor play environments ===
Affordance theory was developed by Gibson (1979) to explain the opportunities for action that environment  provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford  different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being.

Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's emotional well-being. 

'''Table 2'''. ''Examples of environmental features and possible affordances'' 
{| class=&quot;wikitable&quot;
|+
!Environmental feature                             
!Possible affordance for play                                                        
|-
|Fixed structures (e.g. climbing frame)
|Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking
|-
|Natural elements (e.g. tree, rocks, sand)
|Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences
|-
|Floor markings (e.g. painted games)
|Can support structured activities like rule based play, footy, tennis, etc.
|-
|Loose equipment (e.g. balls, tyre, boxes) 
|Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play
|-
|Open play area
|Running, group play,  or flexible use
|}
Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 4 Same space, different opportunities'''
Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. 

* Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. 
* Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play.

Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. 
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Which statement best applies affordance theory to the example?.
|type=&quot;()&quot;}
-A. The play ground causes children to be more physically active.
+B. The same playground can provide different opportunities for play depending on the child and how they use the environment.
-C. Climbing structures always lead to better emotional well-being.
-D. Open spaces are only useful for social play.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
=== '''Social relationship and inclusion''' ===
Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult.  

Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018).

Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018).

=== '''Autonomy, parents and time''' ===
Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being.

Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence.

=== '''Outdoor environments and accessibility''' ===
Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. 

Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025;  Gibson et al., 2017; Pereira et al., 2024). 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |'''Case study: Part 5 Same play space, different experiences'''
{| class=&quot;wikitable&quot;
|Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. 
|}

The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play.
Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence
{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why might two children have different outdoor play experiences even they use the same playground?.
|type=&quot;()&quot;}
+A. Physical, social and personal contributions can affect who child are able to participate.
-B. Outdoor spaces provide the same opportunities for every child.
-C. Accessibility is the only factor that affects outdoor play..
-D. Children who know other children always have better emotional well-being.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''What does recent research show about outdoor play and children's emotional well-being?''' ==
Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being.  

Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play.  

Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being.

=== '''Consistency of recent research findings''' ===
Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes.

Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings.  Lee et al. (2025) found that  more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play.

The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. 

Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect.

=== '''Strengths and limitations of current research''' ===
The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures.

Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors.

Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant.

Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research.

T'''able 3. ''Comparison of recent research'''''

{| class=&quot;wikitable&quot;
|+
!Study               
!Research method             
!Research strength                      
!Limitation                                                
|-
|  &lt;small&gt;Lee et al. (2025)&lt;/small&gt;                                     
|Observational; 6 month follow-up
|Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation.     
|Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings.
|-
|Ferguson et al. (2025)
|Cross sectional; multi ethnic urban cohort (N=2,568)
|Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play.
|Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts.
|-
|Davenport et al. (2024)
|Observational; preschool sample (N=107)
|Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA)
|Small sample and most tested associations with developmental outcomes were not statistically significant.
|-
|Dodd et al. (2026)
|Longitudinal cohort (N = 4,151)
|Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories.
|Relied on parent reported measures, and the observational data cannot not establish conclusions.
|-
|Poulos et al. (2026)
|Recess comparison study (N=317)
|Compared positive and negative affect after outdoor gymnasium and classroom recess
|Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess.
|}
As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. 

Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. 
{| class=&quot;wikitable&quot;
| valign=&quot;top&quot; |{{Robelbox|theme=12|title=Test your learning}}
&lt;quiz display=simple&gt;

{Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being?
|type=&quot;()&quot;}
-A. Because all the studies used different countries.
+B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design.
-C. Because longitudinal studies are always more reliable and accurate.
-D. Because outdoor play only affects positive outcomes.


&lt;/quiz&gt;
{{Robelbox/close}}

|}
== '''How do families, communities and schools support outdoor play opportunities?''' ==
[[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]]
Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools.

=== '''Families''' ===
Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).  

Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors  (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. 

Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. 

Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive.

=== '''Communities''' ===
Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser &amp; van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings.

The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate.

The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being.

=== '''Schools''' ===
Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints.

The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. 

Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. 
* Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier.
* School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints.
* Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess     settings.
* Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess.

{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Children who have access to the same outdoor playground may still have different opportunities to participate in play.
|type=&quot;()&quot;}

+ True
- False

&lt;/quiz&gt;
{{Robelbox/close}}


{{Robelbox|theme=12|title=Quiz}}
&lt;quiz display=simple&gt;

{Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured.
|type=&quot;()&quot;}

- True
+ False

&lt;/quiz&gt;
{{Robelbox/close}}
{| class=&quot;wikitable&quot;
|Scenario

Two primary school aged children A and B enjoy playing  outdoors. They both lived near the same playground. Child A was very familiar  with the playground and knows several other children who play there and  easily join the games. While child B has disability, uses mobility device and  does not know other children at playground. Some parts of the playground are  difficult to access and joining other children who are already playing can  also be challenging. Child B's parents can provide support, although the child  also prefers to play more independently. 

Consider both children can access the same playground, but  do they have same level opportunities to take part in outdoor play?
|}

== '''Conclusion''' ==

* Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared.
* Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant.
* Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}}

https://en.wikipedia.org/wiki/Emotional_dysregulation&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Autonomy&lt;nowiki/&gt;- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

https://en.wikipedia.org/wiki/Longitudinal_study&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

[[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article

https://en.wikipedia.org/wiki/Affordance&lt;nowiki/&gt;-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}}

==References==
{{Hanging indent|1=

Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., &amp; Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health &amp; Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154

Biino, V., Pesce, C., &amp; Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594

Davenport, C., Kuzik, N., Larouche, R., &amp; Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119

de Lannoy, L., Barbeau, K., Seguin, N., &amp; Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01

Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., &amp; McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175

Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., &amp; McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293

Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., &amp; Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9

Lee, J. J., Flouri, E., &amp; Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health &amp; Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020

Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., &amp; Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606

Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., &amp; Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health &amp; Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235

Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., &amp; Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135

}}

==External links==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

https://www.child-encyclopedia.com/outdoor-play&lt;nowiki/&gt;-Encyclopedia in early childhood development

https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf&lt;nowiki/&gt;-Position Statement

https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more&lt;nowiki/&gt;-World Health Organisation

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Children]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
      <sha1>j8w1f8r8e3eg9ucf2j0m3olklfidu8y</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment</title>
    <ns>1</ns>
    <id>331238</id>
    <revision>
      <id>2834787</id>
      <parentid>2834662</parentid>
      <timestamp>2026-09-28T04:05:49Z</timestamp>
      <contributor>
        <username>StretchBeyond</username>
        <id>3105744</id>
      </contributor>
      <comment>/* Feedback */ Reply</comment>
      <origin>2834787</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12125" sha1="akpajc6p0ab3st5mvjk4csfw3cjj7tp" xml:space="preserve">==Vermetten image==
{{ping|StretchBeyond}} You noted that &quot;(The Vermetten 2025 reading has an excellent flow chat to use if I can figure out how to get it embedded. the journal article is Creative Commons4.0 - need to ask James.&quot;
: If you can download the image, it can then be uploaded to [[commons:]] and embedded on Wikiversity. However, there is a catch. The license is CC-NC (Non-commercial) which restricts re-use. Wiki Commons only accepts media which are unrestricted for re-use (e.g., public domain or creative commons attribution). Unfortunately, this image couldn't, for example, be re-used in private enterprise. You could potentially create your own image inspired by the original, and then you own the copyright to that image and can upload/embed it. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:21, 18 August 2026 (UTC)

:Thank you. Based on what your advised yesterday, I have had an attempt at integrating some journal concepts into a graphic that I generated with Microsoft CoPilot. I have embedded it into my article now and referenced it in the caption, but I can't get the caption to be visual on the front page. I'm still struggling with how to manipulate figures and boxes, as the edits are not intuitive.  [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 08:34, 19 August 2026 (UTC)
: Well done! Hopefully, [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FImmersive_therapy_for_PTSD_treatment&amp;diff=2823045&amp;oldid=2823035 this] did the trick. The images are clickable through to the licensing etc., so those details don't need to be in the caption. After a while it will probably make sense to use &quot;edit source&quot; which gives more editing control. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:51, 19 August 2026 (UTC)
::Thank you. I'll have some questions on this stuff tomorrow.  [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 08:55, 19 August 2026 (UTC)

== Strong topic development  ==

Just leaving some feedback, as this topic development chapter is genuinely excellent. The theory, mechanisms, and research evidence are integrated at a very high level, and the immersive therapy examples are clear and engaging. It’s one of the strongest drafts I’ve come across. The structure and flow are exceptionally strong, especially for a topic development stage. [[User:StudentUC2026|StudentUC2026]] ([[User talk:StudentUC2026|discuss]] • [[Special:Contributions/StudentUC2026|contribs]]) 10:22, 22 August 2026 (UTC)

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]
# Remove links and citations from headings
# This chapter is likely to exceeed the maximum word count, so consider truncating preliminary material
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
|3=
&lt;!-- Overview--&gt;
# Very good
# Consider providing a compelling real-life or fictional scenario; costs could be cited in subsequent sections
# Consider abbreviating the problem description and moving detail into subsequent sections
# Reduce overcapitalisation; APA is a down-style: https://www.google.com.au/search?q=capitalization+site%3Aapastyle.apa.org
&lt;!-- Style --&gt;
# Use 3rd person point of view for focus questions
&lt;!-- Focus questions --&gt;
# Focus questions are aligned with sub-title and top-level headings
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Key points are well developed for each section
# Prelminary material (e.g., &quot;Why is emotional processing important in PTSD?&quot;)  will likely need to be abbreviated or removed to allow the chapter to concentrate on answering &quot;How does it work and what are the effects?&quot; within the maximimum word count
&lt;!-- Theory and research --&gt;
# Promising balance of theory and research

&lt;!-- Conclusion --&gt;
# Conclusion is well underway
# Concentrate on answering the focus questions
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Promising use of scenarios/examples/case studies
# Keep scenarios brief 
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
# Keep quiz questions brief and embed them in the most relevant sections 
&lt;!-- Tables --&gt;
# Promising use of table(s)
# Table 1 is probably overly detailed; consider how to simplify
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## [[Help:Wikitext quick reference|italicisation]]
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Good
## See in-text comment
&lt;!-- External links --&gt;
# External links
## Good
## See in-text comment
|9=
&lt;!-- User page --&gt;
# Excellent
&lt;!-- Description about self --&gt;
# Excellent description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Outstanding – at least three different types of contributions with direct link(s) to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:00, 29 August 2026 (UTC)

== Authors Statement ==
'''Authors Statement'''

''I certify that no part of this assignment has been copied from any other student’s work or from any other source except where due acknowledgement is made in the assignment. No part of this assignment has been written for me by any other person. The material contained in this assignment has not been previously submitted for assessment in any other formal unit of study.'' 

'''Gen AI Use Statement'''

''I have used the following Generative Artiﬁcial Intelligence (GenAI) tools/services in the preparation of this assessment:''

·       '''GenAI service name and version:''' Microsoft Copilot

·       '''Link to the GenAI service:''' https://copilot.microsoft.com/

·       '''Ways this GenAI service was used:''' for initial ideation in response to the topic and to brainstorm and develop a writing strategy for this material. Copilot was also used at the end of the process to compare the advanced draft against the rubric. Copilot was used to identify any gross logic errors or data discrepancies in the argument. Copilot was used to proofread the final reference list for APA 7 compliance after EndNote 21 automated it. https://m365.cloud.microsoft/chat/conversation/0895ffb5-6712-49d4-890d-1198212346ee?auth=2--[[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 09:15, 18 September 2026 (UTC)

·       '''GenAI service name and version:''' University of Canberra Studiosity

·       '''Link to the GenAI service:''' https://studiosity.com/connect/partners/university-of-canberra/student_home

·       '''Ways this GenAI service was used:''' for university-sponsored AI feedback to improve written work prior to submission. This was particularly useful in maintaining logical flow through the paragraph PEEL structure and in some aspects of grammar improvement.

·       '''GenAI service name and version:''' Grammarly Desktop 2026 for Microsoft Word

·       '''Link to the GenAI service:''' https://www.grammarly.com/

'''Ways this GenAI service was used:''' I used an integrated MS Word Grammarly subscription to help edit grammar, writing clarity, and punctuation. The tool also assists with inadvertent plagiarism detection, ensuring that citations are sound. 

''I confirm that I have not used the Generative Artificial Intelligence service in the development of this work for assessment, other than as declared and acknowledged above, and that I have cited and referenced any GenAI content in my assessment submission, applying the relevant referencing style''. ''I understand that providing false or misleading information in this declaration and acknowledgement may constitute a breach of the'' [https://www.canberra.edu.au/about-uc/policy-and-legislation/legislation/rules/University-of-Canberra-Student-Conduct-Rules-2023.pdf University of Canberra (Student Conduct) Rules 2023.] [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 09:05, 16 September 2026 (UTC)

== Feedback ==
Hi Andrew

Thanks for providing feedback on my book chapter. I have read over your chapter and it looks great! Your writing is clear and moves the reader seamlessly through each section. 

One point for improvement I noticed was that the sentence &quot;Integrating exposure therapy with gamified, posture-adaptive cognitive rehabilitation to support physical and emotional recovery (Figure 5) (Selvakumaran, 2025).&quot; seems to be missing a final part. For example, you may like to consider linking this sentence to your previous sentence by finishing it with something like &quot;...may facilitate better cross-cultural applications of the US-centric ''Bravemind''&quot;. 

I also noticed your use of the Wiki style citations at certain points (i.e. [3]). These were helpful to me as a reader but I believe I read in the marking criteria that we are encouraged to use either APA or wiki style citations, not both. Could you perhaps hyperlink text so the reader can see these sources, or include them in your external sources list instead? 

Another minor note is that your figures are well-chosen and make the chapter easy to follow, but they are hard to read in the size they currently are. Could you perhaps enlarge these so the reader can see them in full detail at a glance, rather than having to enlarge them? I've trialed enlarging Figure 6 for you so you can see if you like the feel of it, although I do appreciate it may break the flow of text.  

I added some Wiki links for some terms lay people may not immediately recognise, including olfactory and haptic, to ensure the chapter is accessible to a wide range of audiences. I also added some links in your conclusion that take the reader back to the part of the chapter you cite.

Lastly, I also encourage citing Table 1 in a relevant point of your text to contextualise the table for readers. 

All the best with your final edits! 

Kind regards, Pepper [[User:U3253363|U3253363]] ([[User talk:U3253363|discuss]] • [[Special:Contributions/U3253363|contribs]]) 10:34, 27 September 2026 (UTC)

:Pepper,
:Thank you very much for taking the time to review it. Your suggestions are great and really useful. I reworked the Bravemind paragraph you flagged. It had become disjointed over time. Also removed the three footnote-style citations. I really appreciate you making the internal conclusion links - I haven't been able to figure them out yet.   [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 04:05, 28 September 2026 (UTC)</text>
      <sha1>akpajc6p0ab3st5mvjk4csfw3cjj7tp</sha1>
    </revision>
  </page>
  <page>
    <title>The John Snow Prediabetes Institute/The Maritime Health database design</title>
    <ns>0</ns>
    <id>331243</id>
    <revision>
      <id>2834675</id>
      <parentid>2832402</parentid>
      <timestamp>2026-09-27T14:25:53Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834675</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13442" sha1="azmbn5t4fs34wqitjaydejpmuxeftyn" xml:space="preserve">
-------------

'''Design of the project''' (Informed Consent Form below) 

This project is an international, multicentre, observational longitudinal health database that uses routinely collected maritime health examination data to monitor the prevalence and trends of prediabetes, diabetes T-2, hypertension, prehypertension, overweight, obesity and established prevention among maritime workers and students. The intention is to register and offer standardized prevention Inspired by the “Danish health database” (ref. 1)References 
1.	Health of the Danes  https://www.danskernessundhed.dk/

'''Strengths of the design''' • Uses routinely collected data, making it relatively inexpensive. • Can accumulate large international datasets. • Reflects real-world clinical practice. • Supports long-term surveillance of maritime workers' health. • Allow for evaluation of preventive interventions within routine healthcare. Organisers• The John Snow Prediabetes Institute in collaboration with: • Maritime Medical Health Clinics,  • The International Maritime Health Associtation (IMHA), • the Filippine Maritime Health Association

'''Objectives'''

• To establish an “International maritime health database” with samples of health examination data from the maritime clinics and the maritime schools. • To carry out literature studies on lifestyle intervention in the contexts • To describe and follow the prevalence of the target variables by age and gender over the years. (Prediabetes, T2DM, Blood pressure, BMI etc.) • To identify and offer the prediabetes remission program and to follow the effects.

'''Data content'''

Random samples of data extracted from the routine maritime health examinations in specified calendar periods.

Variables • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer or non-officer on board • Blood pressure • Blood sugar HbA1c or Fasting Glucose, or Post prandial • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Participating in a life-style health promotion plan y/n

Automatic recoding of data Excel-recoding formulas-Dec-1-2024 • Blood pressure: Normal, prehypertension, stage 1, 2 • Blood sugar: Normal, prediabetes, diabetes mellitus type 2, • BMI: normal, overweight, obese

'''Blood-sugar-Measurements'''

Either A1c mmol/mol or Fasting mg/dl (glucometer) or Postprandial sugar: mg/dl (glucometer) Validity of the method. (ref 2) Clinics without all the variables are invited to give the data they have.

'''Use of the data for prevention'''

To describe the prevalence of Prediabetes, T2DM, Hypertension, Prehypertension and BMI in the age- and gender groups and occupational groups. Possibly, to offer a prediabetes, prehypertension remission program. To propose (inter) national maritime health policies. To be used for the student’s theses. To publish in international health journals.

Personal prevention, 

'''Type of study •''' International, multicenter observational prevalence database, • Longitudinal surveillance system using repeated routine maritime health prevalence examination data. • Includes an intervention component for individuals identified with prediabetes.

'''Populations sources''' 

The database will include random samples of data extracted from routine health examinations conducted at: • Maritime clinics • Maritime schools Participants may include: • Seafarers • Fishers • Maritime students • Officers and non-officers

'''Data collection''' 

Retrospectively, the study uses secondary data obtained from existing routine medical examinations but can also collect new research-specific measurements prospectively. Key variables include: • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer on board7 non officer • Blood pressure • Blood sugar HbA1c or Fasting Glucose • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Actually, included in a health promotion program 

'''Ethics and data governance Need for ethics review.'''

In a retrospective desgn, formal ethics review is not required, if the study involves fully de-identified pre-existing medical records, with no direct participant contact and no collection of identifiable personal information. In accordance with institutional and national guidelines for minimal-risk research, this type of secondary data analysis is automatically exempt from ethics committee review. 

If the study uses sensitive health data and follows people over time in a prospective design ethics review is needed. Linking records, international data sharing, and testing a prevention programme can create risks, even when names are removed. Before collecting, linking, sharing, or analysing data, each study site must send the study plan and participant information, consent process, data-flow description, and data-management plan to the appropriate ethics committee. Each site needs written approval or a legally permitted exemption. Major changes must be approved before use. If the study involves only non-sensitive health data, the Ethics committees should be involved.

'''Ethical framework and legal compliance.'''

The study will follow the Declaration of Helsinki, national research rules, the EU General Data Protection Regulation (GDPR) where it applies, and the rules of participating countries. Before enrolment or data transfer, the team must record who controls and processes the data and the legal reason for using health information.

'''Consent and participant rights.'''

Participants will receive clear information about the study’s length, health data, follow-up, record linking, international sharing, and approved future research. Consent will be obtained when required. An ethics committee must approve use of data without consent. Refusing or leaving will not affect healthcare, work, training, or education. No new data will be collected after withdrawal, but anonymous data or completed analyses may not be removable. Participants will be told this beforehand.

'''Long-term follow-up and record linking.'''

Each person will receive a study code, allowing records to be linked without putting names in the central database. Each clinic or school will keep the code key separately with restricted access. The central database will hold coded data, or anonymous data when linking is finished. Exact dates and unusual details will be grouped to reduce identification risk. Further contact or record links need legal and ethics approval.

'''Where will data be stored.'''

Data will be kept only on approved servers or encrypted research systems. Names and local code keys will remain at the original site in a separate, encrypted location. Personal computers, unencrypted devices, personal email, and unapproved cloud services are banned. Security includes encryption, backups, logs, malware protection, and an incident plan. Study documents must name the host institution and storage location.

'''Who can access the data.'''

Only named, authorised team members may access the minimum information needed for approved work. They must follow confidentiality rules, use individual accounts and secure sign-in, and lose access when their role ends. Collaborators may receive only necessary coded or anonymous data after approval. Employers, crewing companies, insurers, and other unauthorised parties will not receive personal-level data. Reports will hide small groups or identifying details.

'''Data sharing, retention, and destruction.'''

Institutions will sign agreements covering responsibilities, approved uses, security, breach reporting, further sharing, and data return or deletion. International transfers must follow the law and use extra safeguards when required. Before the study starts, the plan must state how long data will be kept (e.g. 5 years). Afterwards, files and backups will be securely deleted or made permanently anonymous. Code keys will be destroyed unless longer storage is approved.

'''Risk management and oversight.'''

The main risk is that private information could be exposed or linked to a person, causing work, social, or insurance harm. The team will collect only necessary data, separate names from research data, limit access, train staff, secure transfers and storage, keep logs, and respond quickly to suspected breaches. The coordinating researcher will record approvals, access permissions, agreements, and incidents. Any prevention or lifestyle programme tested as research needs ethics approval before it begins.

'''Exclusion Criteria''' 

• Inability to provide informed consent  

• Expressed no-interest to participate 

'''The primary outcomes include''' • Prevalence of Prediabetes, and T2DM • Prevalence of Hypertension and Prehypertension • Prevalence of Overweight and Obesity • Trends by age and sex over time • Prevalence of participation in a prevention program Planned analyses • Estimate disease prevalence. • Compare health indicators across age, gender and working groups. • Monitor trends over multiple years. • Evaluate outcomes of lifestyle interventions. • Support health policy development, student research, and scientific publications.

'''Statistical analysis plan'''

''1. Data preparation''

Check the dataset for missing values, impossible values, and duplicates. Recode the main variables into categories: BMI as normal, overweight, or obese; blood pressure as normal, prehypertension, stage 1, or stage 2; and blood sugar as normal, prediabetes, or type 2 diabetes.

''2. Descriptive statistics''

Summarize age using mean, standard deviation, median, and range. Summarize categorical variables such as gender, work type, smoking, medication use, BMI group, blood pressure group, and blood sugar group using numbers and percentages. These data is presented in the standard -29, 30-49. 50+ age- and gender groups. 

''3. Prevalence estimates in the main age-groups.''

Calculate the prevalence of overweight, obesity, prediabetes, diabetes, prehypertension, and hypertension in the  Report each prevalence as a percentage with a 95% confidence interval ''n the main age-groups -29, 30-49, 50+''

''4. Group comparisons''

Compare prevalence by the 3 age group, gender, work type, and officer/non-officer status. Use chi-square tests for categorical comparisons and t-tests or ANOVA for continuous variables such as BMI or blood pressure.

''5. Trend analysis over time''

f data are available from several years, changes in prevalence over time using tables and line graphs. Test trends using chi-square trend tests or simple logistic regression with year as the predictor.

''6. Prediabetes remission program analysis''

Among participants offered the lifestyle intervention, compare blood sugar, BMI, and blood pressure before and after the program. Use paired t-tests for continuous measures and McNemar’s test for changes in categories, such as prediabetes returning to normal blood sugar.

''7. Missing data''

Report the amount of missing data for each variable. For the simple analysis, use available-case analysis and clearly state the number of participants included in each analysis.

''8. Statistical significance''
Use a significance level of 0.05. Present results with 95% confidence intervals rather than relying only on p-values.Principio del formulario

'''Prevention plan'''

'''Personal prevention''' reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)

'''INFORMED CONSENT FORM'''

International Maritime Health Database 

You are invited to participate in the International Maritime Health Database, which aims to improve the health of seafarers, fishers, and maritime students by monitoring conditions such as prediabetes, type 2 diabetes, hypertension, and obesity. If you agree to participate, information from your routine maritime medical examination (such as age, sex, height, weight, blood pressure, blood glucose, smoking status, medication, and occupation) will be included in a confidential research database. Your participation is voluntary. You may refuse to participate or withdraw your consent at any time without affecting your medical care, employment, or education. Your personal information will be kept confidential. Your data will be coded whenever possible and used only for research, education, and publication in a way that does not identify you. If your results indicate an increased risk of disease, you may be offered information about a voluntary health promotion or prevention programme. By signing below, you confirm that: • You have read and understood this information. • You have had the opportunity to ask questions. • You voluntarily agree to participate in this study. Participant Name: _______________________________ Signature: ______________________________________ Date: __________________________________________ Researcher/Health Professional: ___________________ Signature: ______________________________________ Date: __________________________________________</text>
      <sha1>azmbn5t4fs34wqitjaydejpmuxeftyn</sha1>
    </revision>
    <revision>
      <id>2834676</id>
      <parentid>2834675</parentid>
      <timestamp>2026-09-27T14:27:02Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834676</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12762" sha1="drgj3uogksdfoetr02hjhz8khzmt6rm" xml:space="preserve">
-------------

'''Design of the project''' (Informed Consent Form below) 

This project is an international, multicentre, observational longitudinal health database that uses routinely collected maritime health examination data to monitor the prevalence and trends of prediabetes, diabetes T-2, hypertension, prehypertension, overweight, obesity and established prevention among maritime workers and students. The intention is to register and offer standardized prevention Inspired by the “Danish health database” (ref. 1)References 
1.	Health of the Danes  https://www.danskernessundhed.dk/

'''Strengths of the design''' • Uses routinely collected data, making it relatively inexpensive. • Can accumulate large international datasets. • Reflects real-world clinical practice. • Supports long-term surveillance of maritime workers' health. • Allow for evaluation of preventive interventions within routine healthcare. Organisers• The John Snow Prediabetes Institute in collaboration with: • Maritime Medical Health Clinics,  • The International Maritime Health Associtation (IMHA), • the Filippine Maritime Health Association

'''Objectives'''

• To establish an “International maritime health database” with samples of health examination data from the maritime clinics and the maritime schools. • To carry out literature studies on lifestyle intervention in the contexts • To describe and follow the prevalence of the target variables by age and gender over the years. (Prediabetes, T2DM, Blood pressure, BMI etc.) • To identify and offer the prediabetes remission program and to follow the effects.

'''Data content'''

Random samples of data extracted from the routine maritime health examinations in specified calendar periods.

Variables • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer or non-officer on board • Blood pressure • Blood sugar HbA1c or Fasting Glucose, or Post prandial • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Participating in a life-style health promotion plan y/n

Automatic recoding of data Excel-recoding formulas-Dec-1-2024 • Blood pressure: Normal, prehypertension, stage 1, 2 • Blood sugar: Normal, prediabetes, diabetes mellitus type 2, • BMI: normal, overweight, obese

'''Blood-sugar-Measurements'''

Either A1c mmol/mol or Fasting mg/dl (glucometer) or Postprandial sugar: mg/dl (glucometer) Validity of the method. (ref 2) Clinics without all the variables are invited to give the data they have.

'''Use of the data for prevention'''

To describe the prevalence of Prediabetes, T2DM, Hypertension, Prehypertension and BMI in the age- and gender groups and occupational groups. Possibly, to offer a prediabetes, prehypertension remission program. To propose (inter) national maritime health policies. To be used for the student’s theses. To publish in international health journals.

Personal prevention, 

'''Type of study •''' International, multicenter observational prevalence database, • Longitudinal surveillance system using repeated routine maritime health prevalence examination data. • Includes an intervention component for individuals identified with prediabetes.

'''Populations sources''' 

The database will include random samples of data extracted from routine health examinations conducted at: • Maritime clinics • Maritime schools Participants may include: • Seafarers • Fishers • Maritime students • Officers and non-officers

'''Data collection''' 

Retrospectively, the study uses secondary data obtained from existing routine medical examinations but can also collect new research-specific measurements prospectively. Key variables include: • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer on board7 non officer • Blood pressure • Blood sugar HbA1c or Fasting Glucose • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Actually, included in a health promotion program 

'''Ethics and data governance Need for ethics review.'''

In a retrospective desgn, formal ethics review is not required, if the study involves fully de-identified pre-existing medical records, with no direct participant contact and no collection of identifiable personal information. In accordance with institutional and national guidelines for minimal-risk research, this type of secondary data analysis is automatically exempt from ethics committee review. 

'''Ethical framework and legal compliance.'''

The study will follow the Declaration of Helsinki, national research rules, the EU General Data Protection Regulation (GDPR) where it applies, and the rules of participating countries. Before enrolment or data transfer, the team must record who controls and processes the data and the legal reason for using health information.

'''Consent and participant rights.'''

Participants will receive clear information about the study’s length, health data, follow-up, record linking, international sharing, and approved future research. Consent will be obtained when required. An ethics committee must approve use of data without consent. Refusing or leaving will not affect healthcare, work, training, or education. No new data will be collected after withdrawal, but anonymous data or completed analyses may not be removable. Participants will be told this beforehand.

'''Long-term follow-up and record linking.'''

Each person will receive a study code, allowing records to be linked without putting names in the central database. Each clinic or school will keep the code key separately with restricted access. The central database will hold coded data, or anonymous data when linking is finished. Exact dates and unusual details will be grouped to reduce identification risk. Further contact or record links need legal and ethics approval.

'''Where will data be stored.'''

Data will be kept only on approved servers or encrypted research systems. Names and local code keys will remain at the original site in a separate, encrypted location. Personal computers, unencrypted devices, personal email, and unapproved cloud services are banned. Security includes encryption, backups, logs, malware protection, and an incident plan. Study documents must name the host institution and storage location.

'''Who can access the data.'''

Only named, authorised team members may access the minimum information needed for approved work. They must follow confidentiality rules, use individual accounts and secure sign-in, and lose access when their role ends. Collaborators may receive only necessary coded or anonymous data after approval. Employers, crewing companies, insurers, and other unauthorised parties will not receive personal-level data. Reports will hide small groups or identifying details.

'''Data sharing, retention, and destruction.'''

Institutions will sign agreements covering responsibilities, approved uses, security, breach reporting, further sharing, and data return or deletion. International transfers must follow the law and use extra safeguards when required. Before the study starts, the plan must state how long data will be kept (e.g. 5 years). Afterwards, files and backups will be securely deleted or made permanently anonymous. Code keys will be destroyed unless longer storage is approved.

'''Risk management and oversight.'''

The main risk is that private information could be exposed or linked to a person, causing work, social, or insurance harm. The team will collect only necessary data, separate names from research data, limit access, train staff, secure transfers and storage, keep logs, and respond quickly to suspected breaches. The coordinating researcher will record approvals, access permissions, agreements, and incidents. Any prevention or lifestyle programme tested as research needs ethics approval before it begins.

'''Exclusion Criteria''' 

• Inability to provide informed consent  

• Expressed no-interest to participate 

'''The primary outcomes include''' • Prevalence of Prediabetes, and T2DM • Prevalence of Hypertension and Prehypertension • Prevalence of Overweight and Obesity • Trends by age and sex over time • Prevalence of participation in a prevention program Planned analyses • Estimate disease prevalence. • Compare health indicators across age, gender and working groups. • Monitor trends over multiple years. • Evaluate outcomes of lifestyle interventions. • Support health policy development, student research, and scientific publications.

'''Statistical analysis plan'''

''1. Data preparation''

Check the dataset for missing values, impossible values, and duplicates. Recode the main variables into categories: BMI as normal, overweight, or obese; blood pressure as normal, prehypertension, stage 1, or stage 2; and blood sugar as normal, prediabetes, or type 2 diabetes.

''2. Descriptive statistics''

Summarize age using mean, standard deviation, median, and range. Summarize categorical variables such as gender, work type, smoking, medication use, BMI group, blood pressure group, and blood sugar group using numbers and percentages. These data is presented in the standard -29, 30-49. 50+ age- and gender groups. 

''3. Prevalence estimates in the main age-groups.''

Calculate the prevalence of overweight, obesity, prediabetes, diabetes, prehypertension, and hypertension in the  Report each prevalence as a percentage with a 95% confidence interval ''n the main age-groups -29, 30-49, 50+''

''4. Group comparisons''

Compare prevalence by the 3 age group, gender, work type, and officer/non-officer status. Use chi-square tests for categorical comparisons and t-tests or ANOVA for continuous variables such as BMI or blood pressure.

''5. Trend analysis over time''

f data are available from several years, changes in prevalence over time using tables and line graphs. Test trends using chi-square trend tests or simple logistic regression with year as the predictor.

''6. Prediabetes remission program analysis''

Among participants offered the lifestyle intervention, compare blood sugar, BMI, and blood pressure before and after the program. Use paired t-tests for continuous measures and McNemar’s test for changes in categories, such as prediabetes returning to normal blood sugar.

''7. Missing data''

Report the amount of missing data for each variable. For the simple analysis, use available-case analysis and clearly state the number of participants included in each analysis.

''8. Statistical significance''
Use a significance level of 0.05. Present results with 95% confidence intervals rather than relying only on p-values.Principio del formulario

'''Prevention plan'''

'''Personal prevention''' reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)

'''INFORMED CONSENT FORM'''

International Maritime Health Database 

You are invited to participate in the International Maritime Health Database, which aims to improve the health of seafarers, fishers, and maritime students by monitoring conditions such as prediabetes, type 2 diabetes, hypertension, and obesity. If you agree to participate, information from your routine maritime medical examination (such as age, sex, height, weight, blood pressure, blood glucose, smoking status, medication, and occupation) will be included in a confidential research database. Your participation is voluntary. You may refuse to participate or withdraw your consent at any time without affecting your medical care, employment, or education. Your personal information will be kept confidential. Your data will be coded whenever possible and used only for research, education, and publication in a way that does not identify you. If your results indicate an increased risk of disease, you may be offered information about a voluntary health promotion or prevention programme. By signing below, you confirm that: • You have read and understood this information. • You have had the opportunity to ask questions. • You voluntarily agree to participate in this study. Participant Name: _______________________________ Signature: ______________________________________ Date: __________________________________________ Researcher/Health Professional: ___________________ Signature: ______________________________________ Date: __________________________________________</text>
      <sha1>drgj3uogksdfoetr02hjhz8khzmt6rm</sha1>
    </revision>
    <revision>
      <id>2834677</id>
      <parentid>2834676</parentid>
      <timestamp>2026-09-27T14:28:17Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834677</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12763" sha1="0it88n5noq7zjfytozw2b9pf94cuave" xml:space="preserve">
-------------

'''Design of the project''' (Informed Consent Form below) 

This project is an international, multicentre, observational longitudinal health database that uses routinely collected maritime health examination data to monitor the prevalence and trends of prediabetes, diabetes T-2, hypertension, prehypertension, overweight, obesity and established prevention among maritime workers and students. The intention is to register and offer standardized prevention Inspired by the “Danish health database” (ref. 1)References 
1.	Health of the Danes  https://www.danskernessundhed.dk/

'''Strengths of the design''' • Uses routinely collected data, making it relatively inexpensive. • Can accumulate large international datasets. • Reflects real-world clinical practice. • Supports long-term surveillance of maritime workers' health. • Allow for evaluation of preventive interventions within routine healthcare. Organisers• The John Snow Prediabetes Institute in collaboration with: • Maritime Medical Health Clinics,  • The International Maritime Health Associtation (IMHA), • the Filippine Maritime Health Association

'''Objectives'''

• To establish an “International maritime health database” with samples of health examination data from the maritime clinics and the maritime schools. • To carry out literature studies on lifestyle intervention in the contexts • To describe and follow the prevalence of the target variables by age and gender over the years. (Prediabetes, T2DM, Blood pressure, BMI etc.) • To identify and offer the prediabetes remission program and to follow the effects.

'''Data content'''

Random samples of data extracted from the routine maritime health examinations in specified calendar periods.

Variables • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer or non-officer on board • Blood pressure • Blood sugar HbA1c or Fasting Glucose, or Post prandial • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Participating in a life-style health promotion plan y/n

Automatic recoding of data Excel-recoding formulas-Dec-1-2024 • Blood pressure: Normal, prehypertension, stage 1, 2 • Blood sugar: Normal, prediabetes, diabetes mellitus type 2, • BMI: normal, overweight, obese

'''Blood-sugar-Measurements'''

Either A1c mmol/mol or Fasting mg/dl (glucometer) or Postprandial sugar: mg/dl (glucometer) Validity of the method. (ref 2) Clinics without all the variables are invited to give the data they have.

'''Use of the data for prevention'''

To describe the prevalence of Prediabetes, T2DM, Hypertension, Prehypertension and BMI in the age- and gender groups and occupational groups. Possibly, to offer a prediabetes, prehypertension remission program. To propose (inter) national maritime health policies. To be used for the student’s theses. To publish in international health journals.

Personal prevention, 

'''Type of study •''' International, multicenter observational prevalence database, • Longitudinal surveillance system using repeated routine maritime health prevalence examination data. • Includes an intervention component for individuals identified with prediabetes.

'''Populations sources''' 

The database will include random samples of data extracted from routine health examinations conducted at: • Maritime clinics • Maritime schools Participants may include: • Seafarers • Fishers • Maritime students • Officers and non-officers

'''Data collection''' 

Retrospectively, the study uses secondary data obtained from existing routine medical examinations but can also collect new research-specific measurements prospectively. Key variables include: • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer on board7 non officer • Blood pressure • Blood sugar HbA1c or Fasting Glucose • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Actually, included in a health promotion program 

'''Ethics and data governance Need for ethics review.'''

In a retrospective desgn, formal ethics review is not required, if the study involves fully de-identified pre-existing medical records, with no direct participant contact and no collection of identifiable personal information. In accordance with institutional and national guidelines for minimal-risk research, this type of secondary data analysis is automatically exempt from ethics committee review. 

'''Ethical framework and legal compliance.'''

The study will follow the Declaration of Helsinki, national research rules, the EU General Data Protection Regulation (GDPR) where it applies, and the rules of participating countries. Before enrolment or data transfer, the team must record who controls and processes the data and the legal reason for using health information.

'''Consent and participant rights.'''

Participants will receive clear information about the study’s length, health data, follow-up, record linking, international sharing, and approved future research. Consent will be obtained when required. An ethics committee must approve use of data without consent. Refusing or leaving will not affect healthcare, work, training, or education. No new data will be collected after withdrawal, but anonymous data or completed analyses may not be removable. Participants will be told this beforehand.

'''Long-term follow-up and record linking.'''

Each person will receive a study code, allowing records to be linked without putting names in the central database. Each clinic or school will keep the code key separately with restricted access. The central database will hold coded data, or anonymous data when linking is finished. Exact dates and unusual details will be grouped to reduce identification risk. Further contact or record links need legal and ethics approval.

'''Where will data be stored.'''

Data will be kept only on approved servers or encrypted research systems. Names and local code keys will remain at the original site in a separate, encrypted location. Personal computers, unencrypted devices, personal email, and unapproved cloud services are banned. Security includes encryption, backups, logs, malware protection, and an incident plan. Study documents must name the host institution and storage location.

'''Who can access the data.'''

Only named, authorised team members may access the minimum information needed for approved work. They must follow confidentiality rules, use individual accounts and secure sign-in, and lose access when their role ends. Collaborators may receive only necessary coded or anonymous data after approval. Employers, crewing companies, insurers, and other unauthorised parties will not receive personal-level data. Reports will hide small groups or identifying details.

'''Data sharing, retention, and destruction.'''

Institutions will sign agreements covering responsibilities, approved uses, security, breach reporting, further sharing, and data return or deletion. International transfers must follow the law and use extra safeguards when required. Before the study starts, the plan must state how long data will be kept (e.g. 5 years). Afterwards, files and backups will be securely deleted or made permanently anonymous. Code keys will be destroyed unless longer storage is approved.

'''Risk management and oversight.'''

The main risk is that private information could be exposed or linked to a person, causing work, social, or insurance harm. The team will collect only necessary data, separate names from research data, limit access, train staff, secure transfers and storage, keep logs, and respond quickly to suspected breaches. The coordinating researcher will record approvals, access permissions, agreements, and incidents. Any prevention or lifestyle programme tested as research needs ethics approval before it begins.

'''Exclusion Criteria''' 

• Inability to provide informed consent  

• Expressed no-interest to participate 

'''The primary outcomes include''' • Prevalence of Prediabetes, and T2DM • Prevalence of Hypertension and Prehypertension • Prevalence of Overweight and Obesity • Trends by age and sex over time • Prevalence of participation in a prevention program Planned analyses • Estimate disease prevalence. • Compare health indicators across age, gender and working groups. • Monitor trends over multiple years. • Evaluate outcomes of lifestyle interventions. • Support health policy development, student research, and scientific publications.

'''Statistical analysis plan'''

''1. Data preparation''

Check the dataset for missing values, impossible values, and duplicates. Recode the main variables into categories: BMI as normal, overweight, or obese; blood pressure as normal, prehypertension, stage 1, or stage 2; and blood sugar as normal, prediabetes, or type 2 diabetes.

''2. Descriptive statistics''

Summarize age using mean, standard deviation, median, and range. Summarize categorical variables such as gender, work type, smoking, medication use, BMI group, blood pressure group, and blood sugar group using numbers and percentages. These data is presented in the standard -29, 30-49. 50+ age- and gender groups. 

''3. Prevalence estimates in the main age-groups.''

Calculate the prevalence of overweight, obesity, prediabetes, diabetes, prehypertension, and hypertension in the  Report each prevalence as a percentage with a 95% confidence interval ''n the main age-groups -29, 30-49, 50+''

''4. Group comparisons''

Compare prevalence by the 3 age group, gender, work type, and officer/non-officer status. Use chi-square tests for categorical comparisons and t-tests or ANOVA for continuous variables such as BMI or blood pressure.

''5. Trend analysis over time''

If data are available from several years, changes in prevalence over time using tables and line graphs. Test trends using chi-square trend tests or simple logistic regression with year as the predictor.

''6. Prediabetes remission program analysis''

Among participants offered the lifestyle intervention, compare blood sugar, BMI, and blood pressure before and after the program. Use paired t-tests for continuous measures and McNemar’s test for changes in categories, such as prediabetes returning to normal blood sugar.

''7. Missing data''

Report the amount of missing data for each variable. For the simple analysis, use available-case analysis and clearly state the number of participants included in each analysis.

''8. Statistical significance''
Use a significance level of 0.05. Present results with 95% confidence intervals rather than relying only on p-values.Principio del formulario

'''Prevention plan'''

'''Personal prevention''' reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)

'''INFORMED CONSENT FORM'''

International Maritime Health Database 

You are invited to participate in the International Maritime Health Database, which aims to improve the health of seafarers, fishers, and maritime students by monitoring conditions such as prediabetes, type 2 diabetes, hypertension, and obesity. If you agree to participate, information from your routine maritime medical examination (such as age, sex, height, weight, blood pressure, blood glucose, smoking status, medication, and occupation) will be included in a confidential research database. Your participation is voluntary. You may refuse to participate or withdraw your consent at any time without affecting your medical care, employment, or education. Your personal information will be kept confidential. Your data will be coded whenever possible and used only for research, education, and publication in a way that does not identify you. If your results indicate an increased risk of disease, you may be offered information about a voluntary health promotion or prevention programme. By signing below, you confirm that: • You have read and understood this information. • You have had the opportunity to ask questions. • You voluntarily agree to participate in this study. Participant Name: _______________________________ Signature: ______________________________________ Date: __________________________________________ Researcher/Health Professional: ___________________ Signature: ______________________________________ Date: __________________________________________</text>
      <sha1>0it88n5noq7zjfytozw2b9pf94cuave</sha1>
    </revision>
    <revision>
      <id>2834678</id>
      <parentid>2834677</parentid>
      <timestamp>2026-09-27T14:38:47Z</timestamp>
      <contributor>
        <username>JSINST</username>
        <id>3110286</id>
      </contributor>
      <origin>2834678</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12755" sha1="ojwqlmh1rk62djgh64wo2bze0yxsyfq" xml:space="preserve">
-------------

'''Design of the project''' (Informed Consent Form below) 

This is an international, multicentre, observational longitudinal health database that uses routinely collected maritime health examination data to monitor the prevalence and trends of prediabetes, diabetes T-2, hypertension, prehypertension, overweight, obesity and established prevention among maritime workers and students. The intention is to register and offer standardized prevention Inspired by the “Danish health database” (ref. 1)References 
1.	Health of the Danes  https://www.danskernessundhed.dk/

'''Strengths of the design''' • Uses routinely collected data, making it relatively inexpensive. • Can accumulate large international datasets. • Reflects real-world clinical practice. • Supports long-term surveillance of maritime workers' health. • Allow for evaluation of preventive interventions within routine healthcare. Organisers• The John Snow Prediabetes Institute in collaboration with: • Maritime Medical Health Clinics,  • The International Maritime Health Associtation (IMHA), • the Filippine Maritime Health Association

'''Objectives'''

• To establish an “International maritime health database” with samples of health examination data from the maritime clinics and the maritime schools. • To carry out literature studies on lifestyle intervention in the contexts • To describe and follow the prevalence of the target variables by age and gender over the years. (Prediabetes, T2DM, Blood pressure, BMI etc.) • To identify and offer the prediabetes remission program and to follow the effects.

'''Data content'''

Random samples of data extracted from the routine maritime health examinations in specified calendar periods.

Variables • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer or non-officer on board • Blood pressure • Blood sugar HbA1c or Fasting Glucose, or Post prandial • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Participating in a life-style health promotion plan y/n

Automatic recoding of data Excel-recoding formulas-Dec-1-2024 • Blood pressure: Normal, prehypertension, stage 1, 2 • Blood sugar: Normal, prediabetes, diabetes mellitus type 2, • BMI: normal, overweight, obese

'''Blood-sugar-Measurements'''

Either A1c mmol/mol or Fasting mg/dl (glucometer) or Postprandial sugar: mg/dl (glucometer) Validity of the method. (ref 2) Clinics without all the variables are invited to give the data they have.

'''Use of the data for prevention'''

To describe the prevalence of Prediabetes, T2DM, Hypertension, Prehypertension and BMI in the age- and gender groups and occupational groups. Possibly, to offer a prediabetes, prehypertension remission program. To propose (inter) national maritime health policies. To be used for the student’s theses. To publish in international health journals.

Personal prevention, 

'''Type of study •''' International, multicenter observational prevalence database, • Longitudinal surveillance system using repeated routine maritime health prevalence examination data. • Includes an intervention component for individuals identified with prediabetes.

'''Populations sources''' 

The database will include random samples of data extracted from routine health examinations conducted at: • Maritime clinics • Maritime schools Participants may include: • Seafarers • Fishers • Maritime students • Officers and non-officers

'''Data collection''' 

Retrospectively, the study uses secondary data obtained from existing routine medical examinations but can also collect new research-specific measurements prospectively. Key variables include: • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer on board7 non officer • Blood pressure • Blood sugar HbA1c or Fasting Glucose • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Actually, included in a health promotion program 

'''Ethics and data governance Need for ethics review.'''

In a retrospective desgn, formal ethics review is not required, if the study involves fully de-identified pre-existing medical records, with no direct participant contact and no collection of identifiable personal information. In accordance with institutional and national guidelines for minimal-risk research, this type of secondary data analysis is automatically exempt from ethics committee review. 

'''Ethical framework and legal compliance.'''

The study will follow the Declaration of Helsinki, national research rules, the EU General Data Protection Regulation (GDPR) where it applies, and the rules of participating countries. Before enrolment or data transfer, the team must record who controls and processes the data and the legal reason for using health information.

'''Consent and participant rights.'''

Participants will receive clear information about the study’s length, health data, follow-up, record linking, international sharing, and approved future research. Consent will be obtained when required. An ethics committee must approve use of data without consent. Refusing or leaving will not affect healthcare, work, training, or education. No new data will be collected after withdrawal, but anonymous data or completed analyses may not be removable. Participants will be told this beforehand.

'''Long-term follow-up and record linking.'''

Each person will receive a study code, allowing records to be linked without putting names in the central database. Each clinic or school will keep the code key separately with restricted access. The central database will hold coded data, or anonymous data when linking is finished. Exact dates and unusual details will be grouped to reduce identification risk. Further contact or record links need legal and ethics approval.

'''Where will data be stored.'''

Data will be kept only on approved servers or encrypted research systems. Names and local code keys will remain at the original site in a separate, encrypted location. Personal computers, unencrypted devices, personal email, and unapproved cloud services are banned. Security includes encryption, backups, logs, malware protection, and an incident plan. Study documents must name the host institution and storage location.

'''Who can access the data.'''

Only named, authorised team members may access the minimum information needed for approved work. They must follow confidentiality rules, use individual accounts and secure sign-in, and lose access when their role ends. Collaborators may receive only necessary coded or anonymous data after approval. Employers, crewing companies, insurers, and other unauthorised parties will not receive personal-level data. Reports will hide small groups or identifying details.

'''Data sharing, retention, and destruction.'''

Institutions will sign agreements covering responsibilities, approved uses, security, breach reporting, further sharing, and data return or deletion. International transfers must follow the law and use extra safeguards when required. Before the study starts, the plan must state how long data will be kept (e.g. 5 years). Afterwards, files and backups will be securely deleted or made permanently anonymous. Code keys will be destroyed unless longer storage is approved.

'''Risk management and oversight.'''

The main risk is that private information could be exposed or linked to a person, causing work, social, or insurance harm. The team will collect only necessary data, separate names from research data, limit access, train staff, secure transfers and storage, keep logs, and respond quickly to suspected breaches. The coordinating researcher will record approvals, access permissions, agreements, and incidents. Any prevention or lifestyle programme tested as research needs ethics approval before it begins.

'''Exclusion Criteria''' 

• Inability to provide informed consent  

• Expressed no-interest to participate 

'''The primary outcomes include''' • Prevalence of Prediabetes, and T2DM • Prevalence of Hypertension and Prehypertension • Prevalence of Overweight and Obesity • Trends by age and sex over time • Prevalence of participation in a prevention program Planned analyses • Estimate disease prevalence. • Compare health indicators across age, gender and working groups. • Monitor trends over multiple years. • Evaluate outcomes of lifestyle interventions. • Support health policy development, student research, and scientific publications.

'''Statistical analysis plan'''

''1. Data preparation''

Check the dataset for missing values, impossible values, and duplicates. Recode the main variables into categories: BMI as normal, overweight, or obese; blood pressure as normal, prehypertension, stage 1, or stage 2; and blood sugar as normal, prediabetes, or type 2 diabetes.

''2. Descriptive statistics''

Summarize age using mean, standard deviation, median, and range. Summarize categorical variables such as gender, work type, smoking, medication use, BMI group, blood pressure group, and blood sugar group using numbers and percentages. These data is presented in the standard -29, 30-49. 50+ age- and gender groups. 

''3. Prevalence estimates in the main age-groups.''

Calculate the prevalence of overweight, obesity, prediabetes, diabetes, prehypertension, and hypertension in the  Report each prevalence as a percentage with a 95% confidence interval ''n the main age-groups -29, 30-49, 50+''

''4. Group comparisons''

Compare prevalence by the 3 age group, gender, work type, and officer/non-officer status. Use chi-square tests for categorical comparisons and t-tests or ANOVA for continuous variables such as BMI or blood pressure.

''5. Trend analysis over time''

If data are available from several years, changes in prevalence over time using tables and line graphs. Test trends using chi-square trend tests or simple logistic regression with year as the predictor.

''6. Prediabetes remission program analysis''

Among participants offered the lifestyle intervention, compare blood sugar, BMI, and blood pressure before and after the program. Use paired t-tests for continuous measures and McNemar’s test for changes in categories, such as prediabetes returning to normal blood sugar.

''7. Missing data''

Report the amount of missing data for each variable. For the simple analysis, use available-case analysis and clearly state the number of participants included in each analysis.

''8. Statistical significance''
Use a significance level of 0.05. Present results with 95% confidence intervals rather than relying only on p-values.Principio del formulario

'''Prevention plan'''

'''Personal prevention''' reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association)

'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework)

'''INFORMED CONSENT FORM'''

International Maritime Health Database 

You are invited to participate in the International Maritime Health Database, which aims to improve the health of seafarers, fishers, and maritime students by monitoring conditions such as prediabetes, type 2 diabetes, hypertension, and obesity. If you agree to participate, information from your routine maritime medical examination (such as age, sex, height, weight, blood pressure, blood glucose, smoking status, medication, and occupation) will be included in a confidential research database. Your participation is voluntary. You may refuse to participate or withdraw your consent at any time without affecting your medical care, employment, or education. Your personal information will be kept confidential. Your data will be coded whenever possible and used only for research, education, and publication in a way that does not identify you. If your results indicate an increased risk of disease, you may be offered information about a voluntary health promotion or prevention programme. By signing below, you confirm that: • You have read and understood this information. • You have had the opportunity to ask questions. • You voluntarily agree to participate in this study. Participant Name: _______________________________ Signature: ______________________________________ Date: __________________________________________ Researcher/Health Professional: ___________________ Signature: ______________________________________ Date: __________________________________________</text>
      <sha1>ojwqlmh1rk62djgh64wo2bze0yxsyfq</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Dark empathy</title>
    <ns>0</ns>
    <id>331262</id>
    <revision>
      <id>2834740</id>
      <parentid>2834099</parentid>
      <timestamp>2026-09-28T00:08:09Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>Correcting grammar in focus questions</comment>
      <origin>2834740</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13611" sha1="0evcltpi9ln8m9jvw0erp5wymvcamgc" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas {{ic|APA style uses serial commas}} could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== Dark empathy ==

* [[Empathy]] is a shared experience of feelings and thoughts while dark empathy is understanding those feelings and harnessing against others
* Development of dark empathy '''i'''s when an individual has high levels of dark traits {{ic|What are dark traits?}} and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others

There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings
[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark triad ===

* There are three elements to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380
Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>0evcltpi9ln8m9jvw0erp5wymvcamgc</sha1>
    </revision>
    <revision>
      <id>2834745</id>
      <parentid>2834740</parentid>
      <timestamp>2026-09-28T00:27:49Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Dark empathy */ updating empathy and dark empathy with citiation</comment>
      <origin>2834745</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13742" sha1="kx55kpzyih06c1eyz81f74inzxyyd72" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas {{ic|APA style uses serial commas}} could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== Dark empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy does not have have a natural morality
*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits {{ic|What are dark traits?}} and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others

There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings
[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark triad ===

* There are three elements to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380
Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>kx55kpzyih06c1eyz81f74inzxyyd72</sha1>
    </revision>
    <revision>
      <id>2834746</id>
      <parentid>2834745</parentid>
      <timestamp>2026-09-28T00:29:03Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* References */ adding reference</comment>
      <origin>2834746</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13957" sha1="duxjnpa766n5jc6ladjv59mmwabg5w2" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas {{ic|APA style uses serial commas}} could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== Dark empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy does not have have a natural morality
*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits {{ic|What are dark traits?}} and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others

There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings
[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark triad ===

* There are three elements to the dark triad: 
** Narcissism
** Machiavellianism
** Psychopathy

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>duxjnpa766n5jc6ladjv59mmwabg5w2</sha1>
    </revision>
    <revision>
      <id>2834747</id>
      <parentid>2834746</parentid>
      <timestamp>2026-09-28T00:40:44Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>adding more information to the dark tetrad</comment>
      <origin>2834747</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13994" sha1="93jgvyifs7hclhogo1393gq1qikzl7f" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas {{ic|APA style uses serial commas}} could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== Dark empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy does not have have a natural morality
*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others

There are two types of empathy: 

* Cognitive empathy - the intellectual understanding of another’s emotions and thoughts without feeling their pain
* Affective empathy - physical and emotional sharing of feelings
[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>93jgvyifs7hclhogo1393gq1qikzl7f</sha1>
    </revision>
    <revision>
      <id>2834748</id>
      <parentid>2834747</parentid>
      <timestamp>2026-09-28T00:46:17Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Types of empathy */  updating dotpoints with more context</comment>
      <origin>2834748</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14278" sha1="ducnkhq4bkxt5ai7i0aux0b8lui4pa0" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas {{ic|APA style uses serial commas}} could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== Dark empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy does not have have a natural morality
*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others. Empathy has bright and dark sides as well two types: 
* Affective empathy is physical and emotional sharing of feelings
**This empathy is experiencing with the other person which is found more commonly in bright empathy ()
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This empathy is the understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>ducnkhq4bkxt5ai7i0aux0b8lui4pa0</sha1>
    </revision>
    <revision>
      <id>2834749</id>
      <parentid>2834748</parentid>
      <timestamp>2026-09-28T00:49:09Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Dark empathy */</comment>
      <origin>2834749</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14292" sha1="8dgodtn4hop1qbxxncdijcos73k0pfp" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind and the schemas {{ic|APA style uses serial commas}} could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== The development of empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy does not have have a natural morality
*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others. Empathy has bright and dark sides as well two types: 
* Affective empathy is physical and emotional sharing of feelings
**This empathy is experiencing with the other person which is found more commonly in bright empathy ()
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This empathy is the understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>8dgodtn4hop1qbxxncdijcos73k0pfp</sha1>
    </revision>
    <revision>
      <id>2834750</id>
      <parentid>2834749</parentid>
      <timestamp>2026-09-28T00:50:57Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <origin>2834750</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14257" sha1="r91vu8ytlrxgejo2ycjsbx6vnvjrcbx" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind, and the schemas could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== The development of empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy does not have have a natural morality
*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others. Empathy has bright and dark sides as well two types: 
* Affective empathy is physical and emotional sharing of feelings
**This empathy is experiencing with the other person which is found more commonly in bright empathy ()
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This empathy is the understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>r91vu8ytlrxgejo2ycjsbx6vnvjrcbx</sha1>
    </revision>
    <revision>
      <id>2834755</id>
      <parentid>2834750</parentid>
      <timestamp>2026-09-28T01:11:33Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* The development of empathy */ developing empathy dotpoints</comment>
      <origin>2834755</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14554" sha1="3qo6c2qec0oclryeph9lnbly40o1xlq" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind, and the schemas could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== The development of empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy is not an innate, rather learned and developed. It can be cultivated through techniques to understand other's feelings and foster pro-social behaviour (Abramson, A, 2021)
*Although, empathy does not have have a natural morality which means the individual has to choose how to use their understanding of other's emotions and feelings. 

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others. Empathy has bright and dark sides as well two types: 
* Affective empathy is physical and emotional sharing of feelings
**This empathy is experiencing with the other person which is found more commonly in bright empathy ()
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This empathy is the understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)
* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>3qo6c2qec0oclryeph9lnbly40o1xlq</sha1>
    </revision>
    <revision>
      <id>2834768</id>
      <parentid>2834755</parentid>
      <timestamp>2026-09-28T01:37:47Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* References */</comment>
      <origin>2834768</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14703" sha1="75ccw42p3ccz764ahidn5wp7izzk7w8" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind, and the schemas could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== The development of empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy is not an innate, rather learned and developed. It can be cultivated through techniques to understand other's feelings and foster pro-social behaviour (Abramson, A, 2021)
*Although, empathy does not have have a natural morality which means the individual has to choose how to use their understanding of other's emotions and feelings. 

=== Types of empathy ===

Empathy is a large element of emotional intelligence in an understanding and sharing feeling with others. Empathy has bright and dark sides as well two types: 
* Affective empathy is physical and emotional sharing of feelings
**This empathy is experiencing with the other person which is found more commonly in bright empathy ()
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This empathy is the understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)
* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Abramson, A. (2021, November 1). Cultivating empathy. Monitor on Psychology, 52(8). https://www.apa.org/monitor/2021/11/feature-cultivating-empathy

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>75ccw42p3ccz764ahidn5wp7izzk7w8</sha1>
    </revision>
    <revision>
      <id>2834770</id>
      <parentid>2834768</parentid>
      <timestamp>2026-09-28T01:42:14Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* Types of empathy */ adding emotional intelligence wikiversity link</comment>
      <origin>2834770</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14707" sha1="hhmmqd5wnszo4gf27ewdhbicbzvqvsl" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind, and the schemas could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== The development of empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy is not an innate, rather learned and developed. It can be cultivated through techniques to understand other's feelings and foster pro-social behaviour (Abramson, A, 2021)
*Although, empathy does not have have a natural morality which means the individual has to choose how to use their understanding of other's emotions and feelings. 

=== Types of empathy ===

Empathy is a large element of [[emotional intelligence]] in an understanding and sharing feeling with others. Empathy has bright and dark sides as well two types: 
* Affective empathy is physical and emotional sharing of feelings
**This empathy is experiencing with the other person which is found more commonly in bright empathy ()
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This empathy is the understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)
* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* It is '''affective dissonance'''—having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022) that is a potential indicator of dark empathy
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*it will impact their mental health causing chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*the control and manipulation is exhausting. It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Abramson, A. (2021, November 1). Cultivating empathy. Monitor on Psychology, 52(8). https://www.apa.org/monitor/2021/11/feature-cultivating-empathy

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>hhmmqd5wnszo4gf27ewdhbicbzvqvsl</sha1>
    </revision>
    <revision>
      <id>2834782</id>
      <parentid>2834770</parentid>
      <timestamp>2026-09-28T02:31:08Z</timestamp>
      <contributor>
        <username>U3228742</username>
        <id>3005570</id>
      </contributor>
      <comment>/* The development of empathy */ expanding on dot points</comment>
      <origin>2834782</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14882" sha1="s765xux39ql90sk95m63z8oe5bessq7" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}{{ic|Make it more clear how this connects to dark empathy}}
Imagine your extroverted friend, Nikki, has lost their confidence and is isolating herself. After months of attempting to catch up for coffee, she shows up late and spend most of the time glued to her phone. She apologizes as she sends messages every 10 minutes to let her partner know what she is doing. 
You ask her how she has been and she spends all her time talking about her partner and mentions she is no longer participating in her usual hobbies. 

You have noticed since she met her partner, Fedrick. She rarely goes anywhere without him. She says he has the same interests as her and enjoys spending time with her. After an hour, Fedrick turns up at the cafe to join the conversation. His stories are often focused on himself and his shared trauma with Nikki. 

You are starting to suspect that Fedrick has been manipulating Nikki and your friend seems unhappy despite telling you otherwise. 

{{RoundBoxBottom}}

* Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage
* Understanding why and how people use dark empathy is important to preventing and addressing the the concern
* Research indicates understanding emotional intelligence, the dark triad, theory of mind, and the schemas could assist with understanding the motivation behind dark empathy

{{RoundBoxTop|theme=3}}

;Focus questions

*How do we develop empathy?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals mitigate the negative effects of empathy? 

{{RoundBoxBottom}}

== The development of empathy ==

* [[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other determination (Håkansson Eklund &amp; Summer Meranius, 2021). 
*Empathy is not an innate, rather learned and developed. It can be cultivated through techniques to understand other's feelings and foster pro-social behaviour (Abramson, A, 2021)
*Although, empathy does not have have a natural morality which means the individual has to choose how to use their understanding of other's emotions and feelings. 

=== Types of empathy ===

Empathy is a large element of [[emotional intelligence]] in an understanding and sharing feeling with others. Empathy has bright and dark sides as well two types: 
* Affective empathy is physical and emotional sharing of feelings
**This empathy is experiencing with the other person which is found more commonly in bright empathy ()
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This empathy is the understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

[[File:The Dark Triad Traits.svg|thumb|'''Figure 1.''' The dark triad and subsets of the three elements ]]

=== The dark tetrad ===

* There are four elements to the dark tetrad: 
** Narcissism
** Machiavellianism
** Psychopathy
** Sadism

=== Making of a dark empath ===

*Dark Empathy is understanding those feelings and harnessing against others
* Development of dark empathy is when an individual has high levels of dark traits (narcissism, machiavellianism, psychopathy and sadism) and alongside retained or high cognitive and affective empathy capacity (Gojković et al., 2022)
* Dark empath isn't a psychological diagnosis, yet a term how an individual differences from other diagnosis from the dark triad
* Dark empathy is found in individuals with high dark triad traits and cognitive empathy. Cognitive empathy rather than affective empathy allows the individual to understanding emotions and thoughts without having to feel the emotions which in turn makes it easier to manipulate others
* '''An individual with affective dissonance is another indicator of dark empathy. It is''' having twisted or contradictory emotional reactions, such as feeling happy when someone else is suffering that is present in those with high dark traits (Gojković et al., 2022).
*Research demonstrates experiencing trauma as a child where you had to predict and understand other's emotion to keep safe may lead to the develop of dark empathy

== Consequences of dark empathy ==

*The person using dark empathy and those they are manipulating are both impacted by the behaviour and will harm their wellbeing
*While the use of empathy may bring people together through a shared emotion and experience, dark empathy isolates from community

=== Effects on victims of dark empathy ===

*The immediate impacts are startle and unlikely to be noticed, yet over time an individual will develop chronic anxiety, lower self-esteem and emotional dependency

=== Outcome for dark empaths ===

*An individual using dark empathy will find the control and manipulation effective and after time keeping up this level of deception is exhausting. 
*It will impact their mental health potentially leading to depression, mood instability, difficulty maintaining long-term relationships and identity confusion

== Address dark empathy ==

*Dark empathy is not easily recognised in others
*A narcissist or sociopath may use understanding of schemas to influence others while a dark empath uses their empathy to control and most likely enjoy other's vulnerable emotions

=== Regaining themselves: recovering from a dark empathy ===

*Recognise the patterns used by the dark empath
*Prioritise your self and trust your instincts
*Set healthy emotional boundaries in this relationship
*Seek therapy particularly CBT
*Seperation from the dark empath

=== Unravelling from dark empathy ===

*Recognising and acknowledging the pattern of behaviour of controlling those around them and the impacts on theirs and other’s wellbeing
*Attending therapy is the best way to address and understand the behaviours. Some therapies that would be appropriate and effective for dark empaths are:
**Trauma-informed therapy
**[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**[[w:Schema_therapy|Schema therapy]]
**Emotional regulation and accountability training

== Conclusion ==

* Emotional intelligence does not come with morality
* The dark triad is an indicator of dark empathy if individual have high dark traits and affective cognitive empathy
* Cognitive empathy and trauma patterns can develop the drive for using dark empathy
* Effective identification and therapy will assist with managing an individual using empathy against others

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Abramson, A. (2021, November 1). Cultivating empathy. Monitor on Psychology, 52(8). https://www.apa.org/monitor/2021/11/feature-cultivating-empathy

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380

Håkansson Eklund, J., &amp; Summer Meranius, M. (2021). Toward a consensus on the nature of empathy: A review of reviews. Patient education and counseling, 104(2), 300-307. https://doi.org/10.1016/j.pec.2020.08.022 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mikkelsen, E. N., Geraldi, J., &amp; Clegg, S. (2026). The Dark Side of Empathy and Organization Studies. Organization studies. https://doi.org/10.1177/01708406261459079

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/au/basics/dark-triad Dark Triad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Empathy]]</text>
      <sha1>s765xux39ql90sk95m63z8oe5bessq7</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Socioemotional selectivity theory and wellbeing in ageing</title>
    <ns>0</ns>
    <id>331416</id>
    <revision>
      <id>2834751</id>
      <parentid>2834644</parentid>
      <timestamp>2026-09-28T00:52:47Z</timestamp>
      <contributor>
        <username>U3253354</username>
        <id>3109187</id>
      </contributor>
      <origin>2834751</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19081" sha1="5qkexkfikeqc0t5knyevlb1ghohg31x" xml:space="preserve">{{title|Socioemotional selectivity theory and wellbeing in ageing:&lt;br&gt;How do social and emotional experiences affect wellbeing as people age?}}

==Overview==
{{RoundBoxTop|theme=3}}Scenario
Imagine two adults at different stages of life deciding how to spend their weekend. A younger adult chooses to attend a large social event to meet new people and form new connections, while an older adult chooses to spend time with a smaller group of close family and long-term friends. Although the older adult may have a smaller social network, these relationships may provide greater emotional meaning and satisfaction (see figure 1)


{{RoundBoxBottom}}'''Key question:''' Why do social priorities change as people age, and how might these changes influence emotional wellbeing?

Socioemotional selectivity theory (SST) provides one explanation for why social and emotional priorities may change across adulthood. SST proposes that people's perceptions of how much time they have remaining influence the goals they prioritise. When future time is perceived as relatively indefinite, long-lasting goals tend to receive greater priority. When future time is perceived as more limited, emotionally meaningful and immediate goals become relatively more important (Carstensen et al., 1999; Carstensen, 2006).

These motivational changes may help explain age-related differences in social relationships and emotional experiences. Research has found age-related differences in everyday emotional experience, including greater emotional stability with age (Carstensen et al., 2011). While a meta- analysis found evidence of an age-related positivity effect in attention and memory, although its strength varied across study conditions (Reed et al., 2014). Importantly these findings do not mean that ageing automatically leads to better emotional wellbeing. Instead, social relationships, emotional processes, individual differences, and context all need to be considered when examining wellbeing across adulthood. 

{{RoundBoxTop}}
'''Focus questions'''

{{ic|Use a numbered list as shown in Tutorial 2}}

# What is socioemotional selectivity theory, and how does perceived future time influence motivation?
# How do social relationships and priorities change across adulthood?
# How does emotional experience change with age?
# How can social and emotional changes influence wellbeing in older adulthood?
# What are the limitations of socioemotional selectivity theory in explaining age related differences in wellbeing?

{{RoundBoxBottom}}

== '''Socioemotional selectivity and future time perspective''' ==
Socioemotional selectivity theory (SST) proposes that perceptions of future time influence motivation and the goals people prioritise (Carstensen et al., 1999). When the future is perceived as relatively open-ended, people are more likely to prioritise goals such as learning new information, exploring new opportunities, and developing relationships that may provide future benefits. In contrast, when future time is perceived as more limited, emotionally meaningful and present goals become relatively more important (Carstensen et al., 1999; Carstensen, 2006).

This shift in priorities may influence the social choices people make. When future time is perceived as limited, people may become more selective about how and with who they spend their time, placing greater priority on established relationships that provide emotional meaning. From an SST perspective, changes in social relationships across adulthood may therefore reflect changing motivational priorities rather than simply a loss of interest in social interaction (Carstensen et al., 1999).

Although ageing is often associated with a more limited perception of future time, SST proposes that perceived future time, rather than chronological age alone, plays an important role in shaping social and emotional goals (Carstensen et al., 1999). This distinction is important because it suggests that age-related changes in social priorities may partly reflect changes in how individuals perceive their remaining time. SST therefore provided a theoretical framework for understanding why social priorities may change across adulthood and how these changes could influence emotional experiences and wellbeing.

== '''Changing social relationships across adulthood''' ==
Social networks change across adulthood. English and Carstensen (2014) found that social networks tended to increase during young adulthood before becoming smaller across later adulthood. Importantly, this reduction occurred primarily in more peripheral social relationships, while emotionally close relationships remained relatively stable. This suggests that smaller social networks in later adulthood do not necessarily reflect the loss of close relationships.

This pattern of selective narrowing is consistent with SST. Maintaining close relationships while reducing more peripheral relationships may reflect increasing selectivity in how people use their social time as emotionally meaningful goals become relatively more important (Carstensen et al., 1999).

English and Carstensen (2014) also found that older adults reported more positive and less negative emotional responses to members of their social networks, and that the emotional tone of social networks was associated with everyday emotional experience. These findings suggest that the emotional quality of social relationships may be important when considering wellbeing across adulthood. However, having a smaller social network should not automatically be interpreted as producing greater emotional wellbeing. Instead, the findings suggest that the types and emotional quality of relationships people maintain may be important for understanding social and emotional experiences in later adulthood.

== '''Emotional experience and ageing''' ==
Emotional wellbeing does not necessarily decline with age. Carstensen et al. (2011) examined everyday emotional experiences across adulthood using an experience-sampling method, in which participants reported their emotions at randomly selected times during their daily lives. The study followed participants across multiple measurement periods over approximately 10 years, allowing the researchers to examine both age differences and changes in emotional experience over time.

Carstensen et al. (2011) found that overall emotional wellbeing generally became more positive with age. However, this improvement was not unlimited, with evidence suggesting that gains levelled off in later adulthood. Emotional experiences also became more stable with age, meaning that participants showed less fluctuation in their emotional experiences across everyday life. These findings remained after the researchers accounted for several other factors that could be related to emotional experience, including personality, physical health, verbal fluency, and demographic characteristics.

Emotional experience also became more complex with age. Carstensen et al. (2011) found greater co-occurrence of positive and negative emotions as people grew older, meaning that positive and negative emotions were increasingly likely to be experienced together. The researchers described this mixed emotional experience as ''poignancy''. This finding suggests that improved emotional wellbeing in later adulthood should not be interpreted as older adults simply experiencing more positive and fewer negative emotions.

These findings are broadly consistent with SST because motivational priorities may help explain age-related differences in emotional experience (Carstensen et al., 1999). However, the findings do not demonstrate that ageing itself directly causes better emotional wellbeing. Instead, they provide evidence of age-related changes in several aspects of everyday emotional experience and support examining motivational changes as one possible explanation for these patterns.

== '''The age-related positivity effect''' ==
The age-related positivity effect refers to a relative preference among older adults, compared with younger adults, for positive over negative information in attention and memory (Reed et al., 2014). From an SST perspective, changing motivational priorities may influence how emotional information is attended to and remembered.

Reed et al. (2014) conducted a meta-analysis of 100 empirical studies involving 7,129 participants to examine the reliability of the positivity effect. Overall, the researchers found reliable evidence of an age-related positivity effect. Across the studies, older adults showed a greater preference for positive over negative information, whereas younger adults showed a greater preference for negative over positive information. These findings provide evidence that age-related differences in the processing of emotional information occur across a substantial body of research.

However, the strength of the positivity effect varied across study conditions. Reed et al. (2014) found that the effect was larger when cognitive processing was relatively unconstrained and when studies compared groups with wider age differences. When cognitive processing was constrained by the study task, the positivity effect was smaller. This suggests that the positivity effect is not equally strong across all situations and may depend partly on the conditions under which emotional information is processed.

Overall, the findings are consistent with the idea that motivational changes proposed by SST may contribute to age-related differences in attention and memory for emotional information. However, the meta-analysis does not establish that SST-related motivational changes are the sole cause of the positivity effect. Instead, it demonstrates that the effect is reliable overall while also showing that its strength varies according to methodological and sample characteristics (Reed et al., 2014).

== '''Socioemotional selectivity, emotional regulation, and wellbeing''' ==
Taken together, the research suggests that age-related changes in social and emotional priorities may have important implications for wellbeing. Selective maintenance of close relationships (English &amp; Carstensen, 2014), changes in everyday emotional experience (Carstensen et al., 2011), and age-related differences in attention and memory for emotional information (Reed et al., 2014) are broadly consistent with aspects of SST. Together, these findings suggest that motivational priorities, social relationships, and emotional processes may each contribute to understanding wellbeing across adulthood.

However, more positive emotional experiences at older ages should not be interpreted as evidence that older adults are universally better at regulating their emotions. Isaacowitz (2022) reviewed research on ageing and emotion regulation and concluded that existing evidence does not consistently demonstrate an age-related advantage in emotion regulation. Age differences may depend on factors such as the regulation strategy being examined, characteristics of the situation, and how emotion regulation is measured. Therefore, age-related differences in emotional wellbeing cannot simply be explained by assuming that older adults possess superior emotion-regulation abilities.

Overall, SST provides a framework for connecting perceived future time and motivational priorities with social and emotional experiences that may be relevant to wellbeing. However, these relationships are complex, and patterns observed across age groups do not necessarily apply equally to every individual. Understanding wellbeing in later adulthood therefore requires consideration of both the motivational processes proposed by SST and the individual and contextual factors that may influence social and emotional experience.

== '''Limitations of socioemotional selectivity theory''' ==
Although socioemotional selectivity theory provides a useful framework for understanding age-related changes in social and emotional priorities, it does not fully explain all differences in emotional experience across adulthood. SST proposes that perceived future time influences motivational priorities (Carstensen et al., 1999), but evidence for some of the emotional patterns associated with ageing varies according to the conditions in which they are examined. For example, the age-related positivity effect is reliable overall, but its strength varies according to methodological and sample characteristics (Reed et al., 2014). This suggests that age-related emotional patterns cannot be explained by motivational priorities alone.

There is also limited evidence that older adults are consistently better at regulating their emotions. Isaacowitz (2022) found that laboratory and everyday experience-sampling research does not provide consistent evidence of an age-related advantage in emotion regulation. The review also highlights the importance of considering individual differences and context when examining changes in emotion regulation across adulthood. Therefore, although SST offers an important explanation for how perceptions of time may shape social and emotional priorities, it should not be interpreted as a complete explanation of emotional wellbeing in later adulthood (Isaacowitz, 2022).

==Figures==
{| class=&quot;wikitable&quot;
|+
!Perceived future time
!Goals receiving relatively greater priority 
|-
|More open-ended
|Learning, exploring, new opportunities, and relationships that may provide future benefits
|-
|More limited
|Present and emotionally meaningful goals and relationships
|}
[[File:Representaciones de los adultos mayores.jpg|thumb|270x270px|Figure 1. Older adults spending time together. Socioemotional selectivity theory proposes that emotionally meaningful relationships may receive greater priority when future time is perceived as limited.]]


==Learning features==


{{RoundBoxTop|theme=}}{{RoundBoxTop|theme=3}}

'''Case Study: Changing social priorities'''

Maria is 65 and increasingly prefers spending time with close friends and family rather than attending large social gatherings.
Questions
#How could SST explain Maria's changing social preferences?
#Does Maria's smaller social network necessarily indicate greater emotional wellbeing? Why or why not?{{RoundBoxBottom}}

{{RoundBoxTop|theme=3}}
'''Test your knowledge'''

'''1. According to SST, what is most important for understanding changes in motivational priorities?'''

A. Chronological age alone&lt;br&gt;
B. Perceived future time&lt;br&gt;
C. Number of social relationships&lt;br&gt;
D. Personality alone

'''Answer:''' B. Perceived future time

'''2. What does research suggest commonly happens to social networks across later adulthood?'''

A. All relationships decline equally&lt;br&gt;
B. Close relationships disappear first&lt;br&gt;
C. More peripheral relationships tend to decline while close relationships remain relatively stable&lt;br&gt;
D. Social networks always become larger

'''Answer:''' C. More peripheral relationships tend to decline while close relationships remain relatively stable

'''3. Which statement best reflects the evidence discussed in this chapter?'''

A. Older adults are always better at regulating their emotions&lt;br&gt;
B. Ageing automatically improves emotional wellbeing&lt;br&gt;
C. The positivity effect is equally strong in all situations&lt;br&gt;
D. Age-related social and emotional patterns can vary across individuals and situations

'''Answer:''' D. Age-related social and emotional patterns can vary across individuals and situations

{{RoundBoxBottom}}
==Conclusion==
Socioemotional selectivity theory provides a framework for understanding how changing perceptions of future time may influence social and emotional priorities across adulthood. As future time is perceived as more limited, emotionally meaningful goals and relationships tend to become relatively more important (Carstensen et al., 1999). Consistent with this framework, research suggests that social networks may become more selective across adulthood, with close relationships remaining relatively stable while more minor relationships decline (English &amp; Carstensen, 2014).

Emotional wellbeing also does not necessarily decline with age. Research indicates age-related changes in everyday emotional experience, including greater emotional stability, as well as differences in attention and memory for positive and negative information (Carstensen et al., 2011; Reed et al., 2014). However, these patterns do not mean that ageing automatically leads to better emotional regulation or greater wellbeing. Instead, the evidence suggests that social relationships, motivational priorities, emotional process, and individual and contextual factors all need to be considered when understanding wellbeing across adulthood (Isaacowitz, 2022).

Overall, SST helps explain how changing social and emotional priorities may be relevant to wellbeing in later adulthood, while recognising that chronological age alone does not determine emotional wellbeing.

==See also==
** [https://en.wikipedia.org/wiki/Socioemotional_selectivity_theory? Socioemotional selectivity theory] (Wikipedia)
** [[Emotional self-regulation|Emotion]] (Wikiversity)

==References==
{{Hanging indent|1=

Carstensen, L. L. (2006). The influence of a sense of time on human development. ''Science, 312''(5782), 1913–1915. https://doi.org/10.1126/science.1127488 

Carstensen, L. L., Isaacowitz, D. M., &amp; Charles, S. T. (1999). Taking time seriously: A theory of socioemotional selectivity. ''American Psychologist, 54''(3), 165–181. https://doi.org/10.1037/0003-066X.54.3.165

Carstensen, L. L., Turan, B., Scheibe, S., Ram, N., Ersner-Hershfield, H., Samanez-Larkin, G. R., Brooks, K. P., &amp; Nesselroade, J. R. (2011). Emotional experience improves with age: Evidence based on over 10 years of experience sampling. ''Psychology and Aging, 26''(1), 21–33. https://doi.org/10.1037/a0021285

English, T., &amp; Carstensen, L. L. (2014). Selective narrowing of social networks across adulthood is associated with improved emotional experience in daily life. ''International Journal of Behavioral Development'', ''38''(2), 195–202. https://doi.org/10.1177/0165025413515404 

Isaacowitz, D. M. (2022). What do we know about aging and emotion regulation? ''Perspectives on Psychological Science'', ''17''(6), 174569162110598.  https://doi.org/10.1177/17456916211059819

Reed, A. E., Chan, L., &amp; Mikels, J. A. (2014). Meta-analysis of the age-related positivity effect: Age differences in preferences for positive over negative information. ''Psychology and Aging, 29''(1), 1–15. https://doi.org/10.1037/a0035194

}}

==External links==

* [https://www.apa.org/news/press/releases/2019/11/older-adults-social-networks American Psychological Association - Older adults and social wellbeing] 
* [https://lifespan.stanford.edu/research?utm_source Stanford Life-span Development Laboratory – Socioemotional selectivity theory] 

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Ageing]]
[[Category:Motivation and emotion/Book/Emotion]]
[[Category:Motivation and emotion/Book/Social]]</text>
      <sha1>5qkexkfikeqc0t5knyevlb1ghohg31x</sha1>
    </revision>
  </page>
  <page>
    <title>Plant Divisions (Phyla)/Ginkgophyta</title>
    <ns>0</ns>
    <id>331908</id>
    <revision>
      <id>2834706</id>
      <parentid>2834487</parentid>
      <timestamp>2026-09-27T20:09:28Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <origin>2834706</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3885" sha1="joi0gqbuhjdl4cjwdvdk6mi8xdufri2" xml:space="preserve">[[Image:GINKGOBAUM-2.jpg|thumb|300px|right|This is a ''ginkgo''.]]
Ginkgos are a class/phylum/divison of gymnosperms defined by Sergei V. Meyen in 1984 to encompass Ginkgoales (which contains the living Ginkgo) alongside a number of extinct seed plant groups, which he considered to be closely related based on similarities of morphology of pollen, seeds, cuticles, short shoots and leaves.
==Information==
Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct
==Evolutionary history==
Ginkgophyta and [[Plant_Divisions_(Phyla)/Cycadophyta|Cycadophyta]] have a very ancient divergence dating to the Mississippian.&lt;ref&gt;{{Cite journal|last1=Stull|first1=Gregory W.|last2=Qu|first2=Xiao-Jian|last3=Parins-Fukuchi|first3=Caroline|last4=Yang|first4=Ying-Ying|last5=Yang|first5=Jun-Bo|last6=Yang|first6=Zhi-Yun|last7=Hu|first7=Yi|last8=Ma|first8=Hong|last9=Soltis|first9=Pamela S.|last10=Soltis|first10=Douglas E.|last11=Li|first11=De-Zhu|date=19 July 2021|title=Gene duplications and phylogenomic conflict underlie major pulses of phenotypic evolution in gymnosperms|url=https://www.nature.com/articles/s41477-021-00964-4|journal=Nature Plants|language=en|volume=7|issue=8|pages=1015–1025|doi=10.1038/s41477-021-00964-4|pmid=34282286 |bibcode=2021NatPl...7.1015S |s2cid=236141481 |issn=2055-0278|url-access=subscription}}&lt;/ref&gt; The earliest representative of the group in the fossil record is probably ''Trichopitys'' from the Asselian (299-293 million years ago) of France. Alongside other, related forms such as ''Yimaia'' and ''Karkenia'', which have differently arranged reproductive structures and seeds associated with ''Ginkgo''-like leaves, the earliest representatives of ''Ginkgo'', represented by reproductive organs similar to the living species, first appear in the Middle Jurassic, &lt;ref&gt;{{Cite journal|last=Zhou|first=Zhi-Yan|date=March 2009|title=An overview of fossil Ginkgoales|url=https://linkinghub.elsevier.com/retrieve/pii/S1871174X0900002X|journal=Palaeoworld|language=en|volume=18|issue=1|pages=1–22|doi=10.1016/j.palwor.2009.01.001|url-access=subscription}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite book|title=Paleobotany, Second Edition: The Biology and Evolution of Fossil Plants|last1=Taylor|first1=Thomas N.|last2=Taylor|first2=Edith L.|last3=Krings|first3=Michael|date=29 December 2008|publisher=Academic Press|isbn=9780123739728|edition=2nd|language=en}}&lt;/ref&gt; The diversity of Ginkgoales declined during the Late Cretaceous and Cenozoic, coincident with the rise of [[Plant Divisions (Phyla)/Magnoliophyta|Magnoliophytes]], with all Ginkgophytes aside from ''Ginkgo'' being extinct by the end of the Cretaceous.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|url=http://accessscience.com/content/289700|title=Ginkgoales|last=Beck|first=Charles|date=2014|website=Access Science|doi=10.1036/1097-8542.289700 |access-date=13 April 2017|url-access=subscription}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot; /&gt; The only remaining Ginkgophyte was ''Ginkgo adiantoides'' – a polymorphic species.&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=http://www.ucmp.berkeley.edu/seedplants/ginkgoales/ginkgo.html|title=Introduction to the Ginkgoales|author=Jalalpour, Julie|author2=Malkin, Matt|author3=Poon, Peter|author4=Rehrmann, Liz|author5=Yu, Jerry|date=1997|website=www.ucmp.berkeley.edu|access-date=20 April 2017}}&lt;/ref&gt; Modern ''Ginkgo'' trees are native to China.&lt;ref name=&quot;:1&quot; /&gt;
==Orders==
†Czekanowskiales – Ginkgoales
===Questionable Orders===
†Caytoniales? – †Calamopityales? – †Corystospermales? – 
†Peltaspermales? 
†Dicranophyllales?
†Calamopityales?
†Callistophytales?
†Glossopteridales?
†&quot;Palaeophyllales&quot;?

==References==
[[Wikipedia:Ginkgoopsida]]
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>joi0gqbuhjdl4cjwdvdk6mi8xdufri2</sha1>
    </revision>
  </page>
  <page>
    <title>User:The Citer/sandbox</title>
    <ns>2</ns>
    <id>332158</id>
    <revision>
      <id>2834736</id>
      <parentid>2834571</parentid>
      <timestamp>2026-09-27T23:35:56Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <origin>2834736</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="217" sha1="42kcel3t1ptj5ff511h0reg13e8nwsc" xml:space="preserve">My template and text storage!
==Templates==
[[File:|thumb|300x300px|[Insert_text_here]]]
==Text==
–
==Purpose==
It's a storage space so I don't have to copy text in source editor; I just can copy it in reading mode.</text>
      <sha1>42kcel3t1ptj5ff511h0reg13e8nwsc</sha1>
    </revision>
  </page>
  <page>
    <title>Plant Divisions (Phyla)/Pinophyta (Coniferophyta)</title>
    <ns>0</ns>
    <id>332221</id>
    <revision>
      <id>2834757</id>
      <parentid>2834421</parentid>
      <timestamp>2026-09-28T01:17:32Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Added Classes!</comment>
      <origin>2834757</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5250" sha1="g7vp2tu8zsixdm14hrfxuzofnc0m9pa" xml:space="preserve">[[File:Scots Pine (Pinus sylvestris) - Kristiansand, Norway 2021-08-10.jpg|thumb|300x300px|This is a Conifer.]]
Conifers (/ˈkɒnɪfər/) are a group of vascular plants and a subset of gymnosperms. They are woody trees and shrubs, mostly evergreen with a regular branching pattern, reproducing with male and female cones, usually on the same tree. They're wind-pollinated and the seeds are usually dispersed by the wind. Taxonomically, they make up the division Pinophyta, also known as Coniferae. All extant conifers, except for the [https://en.wikiversity.org/w/index.php?title=Gnetophyta&amp;action=edit&amp;redlink=1 gnetophytes], are perennials with secondary growth. There are 629 Extant species.
==Information==
Name Meaning: Pinus-like plant, cone-bearing plant

English Common Name: Conifers

Major distinguishing characteristics: Cones containing seeds and wood composed of tracheids

==Evolutionary History==
The earliest conifers appear in the fossil record during the Late [[Wikipedia:Carboniferous|Carboniferous]] over 300 million years ago.&lt;ref name=&quot;Leslie Beaulieu 2018&quot;&gt;{{cite journal |last1=Leslie |first1=Andrew B. |last2=Beaulieu |first2=Jeremy |last3=Holman |first3=Garth |last4=Campbell |first4=Christopher S. |last5=Mei |first5=Wenbin |last6=Raubeson |first6=Linda R. |last7=Mathews |first7=Sarah |title=An overview of extant conifer evolution from the perspective of the fossil record |journal=American Journal of Botany |volume=105 |issue=10 |pages=1531–1544 |year=2018 |doi=10.1002/ajb2.1143 |pmid=30157290 |bibcode=2018AmJB..105.1531L |url=https://digitalcommons.lsu.edu/biosci_pubs/2619 }}&lt;/ref&gt; The range of conifers expanded during the [[Wikipedia:Cisuralian|Cisuralian]] to lowlands due to increasing aridity. Conifers were largely unaffected by the [[Wikipedia:Permian–Triassic extinction event|Permian–Triassic extinction event]],&lt;ref&gt;{{Cite journal |last1=Nowak |first1=Hendrik |last2=Schneebeli-Hermann |first2=Elke |last3=Kustatscher |first3=Evelyn |date=2019-01-23 |title=No mass extinction for land plants at the Permian–Triassic transition |journal=[[Nature Communications]] |volume=10 |issue=1 |page=384 |bibcode=2019NatCo..10..384N |doi=10.1038/s41467-018-07945-w |pmc=6344494 |pmid=30674875 |doi-access=free}}&lt;/ref&gt; and  became dominant land plants in the Mesozoic, until flowering plants took over many ecosystems in the Cretaceous. Because of that, many conifers today are relict species, surviving in a small part of their former ranges.
==Phylogeny==
The cladogram summarizes the group's external phylogeny. The conifers are gymnosperms, sister to a clade consisting of the [[Plant Divisions (Phyla)/Ginkgophyta|ginkgos]] and  [[Plant Divisions (Phyla)/Cycadophyta|cycads]].&lt;ref name=&quot;Leslie appendix&quot;&gt;{{Cite journal |last=Leslie |first=Andrew B. |display-authors=et al. |year=2018 |title=ajb21143-sup-0004-AppendixS4 |url=https://bsapubs.onlinelibrary.wiley.com/action/downloadSupplement?doi=10.1002%2Fajb2.1143&amp;file=ajb21143-sup-0004-AppendixS4.pdf |journal=[[American Journal of Botany]] |volume=105 |issue=9 |pages=1531–1544 |doi=10.1002/ajb2.1143 |pmid=30157290 |bibcode=2018AmJB..105.1531L |s2cid=52120430}}&lt;/ref&gt;&lt;ref name=&quot;Stull matrix&quot;&gt;{{Cite report |url=https://figshare.com/articles/dataset/Gene_duplications_and_genomic_conflict_underlie_major_pulses_of_phenotypic_evolution_in_gymnosperms/14547354 |title=main.dated.supermatrix.tree.T9.tre |last=Stull |first=Gregory W. |year=2021 |publisher=Figshare |doi=10.6084/m9.figshare.14547354.v1 |display-authors=et al.}}&lt;/ref&gt;

{{clade|style=line-height:100%;
|label1='''Pinophyta'''
|sublabel1=(Coniferae)
|1={{clade
|1={{clade
|label1=Pinaceae |sublabel1=pine family
|1=[[File:Abies alba - Köhler–s Medizinal-Pflanzen-001.jpg|60px]]
|label2=Gnetophyta |sublabel2=(3 extant genera)
|2=[[File:Ephedra distachya (female plant in bloom).jpg|60px]]
}}
|2={{clade
|1={{clade
|label1=Araucariaceae |sublabel1=monkey puzzle family
|1=[[File:Araucaria brasiliana SZ138.jpg|60px]]
|label2=Podocarpaceae |sublabel2=podocarps
|2=[[File:Podocarpus macrophyllus SZ134.png|60px]]
}}
|2={{clade
|label1=Sciadopityaceae |sublabel1=umbrella pines
|1=[[File:Sciadopitys verticillata SZ102.jpg|60px]]
|2={{clade
|label1=Cupressaceae |sublabel1=cypress family
|1=[[File:Cupressus sempervirens (cropped).tiff|60px]]
|label2=Taxaceae |sublabel2=yew family (including ''Cephalotaxus'')
|2=[[File:Nouvelle iconographie fourragère (cropped).jpg|60px]]
}}
}}
}}
}}
}}
==Subclasses==
Cupressidae – Pinidae
===Cladistically included but traditionally excluded taxa===
Gnetidae&lt;ref name=&quot;Yang Ferguson 2022&quot;&gt;{{cite journal |last1=Yang |first1=Yong |last2=Ferguson |first2=David Kay |last3=Liu |first3=Bing |last4=Mao |first4=Kang-Shan |last5=Gao |first5=Lian-Ming |last6=Zhang |first6=Shou-Zhou |last7=Wan |first7=Tao |last8=Rushforth |first8=Keith |last9=Zhang |first9=Zhi-Xiang |display-authors=3 |title=Recent advances on phylogenomics of gymnosperms and a new classification |journal=Plant Diversity |volume=44 |issue=4 |date=2022 |pmid=35967253 |pmc=9363647 |doi=10.1016/j.pld.2022.05.003 |pages=340–350 |bibcode=2022PlDiv..44..340Y }}&lt;/ref&gt;
==References==
[[Wikipedia:Conifer]]
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>g7vp2tu8zsixdm14hrfxuzofnc0m9pa</sha1>
    </revision>
  </page>
  <page>
    <title>Streptophytes</title>
    <ns>0</ns>
    <id>332222</id>
    <revision>
      <id>2834703</id>
      <parentid>2834573</parentid>
      <timestamp>2026-09-27T20:01:36Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Faulty citation removed! (Plus, cycad added.)</comment>
      <origin>2834703</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10584" sha1="5s7bfqxi0jmm55v2xle7lsfav4ivbsh" xml:space="preserve">[[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px]]
Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.&lt;ref name=Jeffrey1967&gt;{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt;
==Information==
Name Meaning: Twisted plant

English Common Name: Streptophytes

Major distinguishing characteristics: 

Approximate number of species described: &gt;350,000
==Classification==
The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.&lt;ref&gt;{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}&lt;/ref&gt;{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,&lt;ref name=Jeffrey1967/&gt;&lt;ref name=Adletal2012&gt;{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}&lt;/ref&gt; Streptophytina); others include more groups (e.g., Charophyta&lt;!-- Karol et al. 2009--&gt;&lt;ref name=Adletal2012/&gt; Streptophyta,&lt;ref name=Bremer1985&gt;{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}&lt;/ref&gt;&lt;!-- de Reviers 2002; --&gt;&lt;ref name=Leliaert-2012&gt;{{cite journal  |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}&lt;/ref&gt; Streptobionta&lt;ref&gt;{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}&lt;/ref&gt;); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,&lt;ref name=&quot;Cavalier-Smith1993&quot;&gt;{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}&lt;/ref&gt;&lt;ref name=Leliaert-2012/&gt; Charophyceae{{citation needed|reason=Mattox &amp; Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.&lt;ref name=Liang2020&gt;{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}&lt;/ref&gt;&lt;ref name=Nedelcu2006&gt;{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}&lt;/ref&gt; 
==Phylogeny==
Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.&lt;ref&gt;{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology &amp; Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt; (Warning, this phylogenetic picture needs to be improved (See [[Talk:Streptophytes|talk]]).)

{{clade
  |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]`
|2={{clade
      |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]]
|2={{clade
|1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]]
  |2={{clade
    |1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]]
    |label2=Phragmoplastophyta
    |2={{clade
      |1=Charophyceae (Stoneworts &amp; musk grasses) [[File:CharaFragilis.jpg|60px]]
      |2={{clade
          |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]]
        |2={{clade
        |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]]          |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]]
           }}
         }}
       }}
     }}
   }}
}}}}

==Diversity==
There are many species of Streptophytes. These are just the phyla:
===Anthocerotophytes===
[[File:Dendroceros.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]&lt;br&gt;
Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300

Classes: Anthocerotopsida, Leiosporocerotopsida
===Bryophytes===
[[File:Tionesta-ac-moss2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Bryophyta]]&lt;br&gt;
Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

Classes: Andreaeobryopsida,
Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida

===Chlorophytes===
[[File:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Chlorophyta]]&lt;br&gt;
Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000
Classes: Chlorodendrophyceae, Chlorophyceae, Chloropicophyceae, †Chuariophyceae, Mamiellophyceae, Nephroselmidophyceae, Pedinophyceae, Picocystophyceae, Pyramimonadophyceae, Trebouxiophyceae, Ulvophyceae
==Cycadophytes==
[[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Chlorophyta]]&lt;br&gt;
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

Classes: Cycadopsida
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>5s7bfqxi0jmm55v2xle7lsfav4ivbsh</sha1>
    </revision>
    <revision>
      <id>2834704</id>
      <parentid>2834703</parentid>
      <timestamp>2026-09-27T20:06:23Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <minor />
      <comment>/* Cycadophytes */</comment>
      <origin>2834704</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10587" sha1="6ja6d0o98q5aum3lt09ffdxn2q916xc" xml:space="preserve">[[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px]]
Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.&lt;ref name=Jeffrey1967&gt;{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt;
==Information==
Name Meaning: Twisted plant

English Common Name: Streptophytes

Major distinguishing characteristics: 

Approximate number of species described: &gt;350,000
==Classification==
The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.&lt;ref&gt;{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}&lt;/ref&gt;{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,&lt;ref name=Jeffrey1967/&gt;&lt;ref name=Adletal2012&gt;{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}&lt;/ref&gt; Streptophytina); others include more groups (e.g., Charophyta&lt;!-- Karol et al. 2009--&gt;&lt;ref name=Adletal2012/&gt; Streptophyta,&lt;ref name=Bremer1985&gt;{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}&lt;/ref&gt;&lt;!-- de Reviers 2002; --&gt;&lt;ref name=Leliaert-2012&gt;{{cite journal  |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}&lt;/ref&gt; Streptobionta&lt;ref&gt;{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}&lt;/ref&gt;); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,&lt;ref name=&quot;Cavalier-Smith1993&quot;&gt;{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}&lt;/ref&gt;&lt;ref name=Leliaert-2012/&gt; Charophyceae{{citation needed|reason=Mattox &amp; Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.&lt;ref name=Liang2020&gt;{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}&lt;/ref&gt;&lt;ref name=Nedelcu2006&gt;{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}&lt;/ref&gt; 
==Phylogeny==
Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.&lt;ref&gt;{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology &amp; Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt; (Warning, this phylogenetic picture needs to be improved (See [[Talk:Streptophytes|talk]]).)

{{clade
  |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]`
|2={{clade
      |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]]
|2={{clade
|1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]]
  |2={{clade
    |1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]]
    |label2=Phragmoplastophyta
    |2={{clade
      |1=Charophyceae (Stoneworts &amp; musk grasses) [[File:CharaFragilis.jpg|60px]]
      |2={{clade
          |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]]
        |2={{clade
        |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]]          |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]]
           }}
         }}
       }}
     }}
   }}
}}}}

==Diversity==
There are many species of Streptophytes. These are just the phyla:
===Anthocerotophytes===
[[File:Dendroceros.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]&lt;br&gt;
Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300

Classes: Anthocerotopsida, Leiosporocerotopsida
===Bryophytes===
[[File:Tionesta-ac-moss2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Bryophyta]]&lt;br&gt;
Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

Classes: Andreaeobryopsida,
Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida

===Chlorophytes===
[[File:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Chlorophyta]]&lt;br&gt;
Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000
Classes: Chlorodendrophyceae, Chlorophyceae, Chloropicophyceae, †Chuariophyceae, Mamiellophyceae, Nephroselmidophyceae, Pedinophyceae, Picocystophyceae, Pyramimonadophyceae, Trebouxiophyceae, Ulvophyceae
===Cycadophytes===
[[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Chlorophyta]]&lt;br&gt;
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

Classes: Cycadopsida

==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>6ja6d0o98q5aum3lt09ffdxn2q916xc</sha1>
    </revision>
    <revision>
      <id>2834707</id>
      <parentid>2834704</parentid>
      <timestamp>2026-09-27T20:10:45Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>/* Cycadophytes */</comment>
      <origin>2834707</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10587" sha1="3noajh5bw9wzo4uvx89oe16mm9jt0ah" xml:space="preserve">[[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px]]
Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.&lt;ref name=Jeffrey1967&gt;{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt;
==Information==
Name Meaning: Twisted plant

English Common Name: Streptophytes

Major distinguishing characteristics: 

Approximate number of species described: &gt;350,000
==Classification==
The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.&lt;ref&gt;{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}&lt;/ref&gt;{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,&lt;ref name=Jeffrey1967/&gt;&lt;ref name=Adletal2012&gt;{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}&lt;/ref&gt; Streptophytina); others include more groups (e.g., Charophyta&lt;!-- Karol et al. 2009--&gt;&lt;ref name=Adletal2012/&gt; Streptophyta,&lt;ref name=Bremer1985&gt;{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}&lt;/ref&gt;&lt;!-- de Reviers 2002; --&gt;&lt;ref name=Leliaert-2012&gt;{{cite journal  |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}&lt;/ref&gt; Streptobionta&lt;ref&gt;{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}&lt;/ref&gt;); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,&lt;ref name=&quot;Cavalier-Smith1993&quot;&gt;{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}&lt;/ref&gt;&lt;ref name=Leliaert-2012/&gt; Charophyceae{{citation needed|reason=Mattox &amp; Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.&lt;ref name=Liang2020&gt;{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}&lt;/ref&gt;&lt;ref name=Nedelcu2006&gt;{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}&lt;/ref&gt; 
==Phylogeny==
Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.&lt;ref&gt;{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology &amp; Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt; (Warning, this phylogenetic picture needs to be improved (See [[Talk:Streptophytes|talk]]).)

{{clade
  |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]`
|2={{clade
      |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]]
|2={{clade
|1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]]
  |2={{clade
    |1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]]
    |label2=Phragmoplastophyta
    |2={{clade
      |1=Charophyceae (Stoneworts &amp; musk grasses) [[File:CharaFragilis.jpg|60px]]
      |2={{clade
          |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]]
        |2={{clade
        |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]]          |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]]
           }}
         }}
       }}
     }}
   }}
}}}}

==Diversity==
There are many species of Streptophytes. These are just the phyla:
===Anthocerotophytes===
[[File:Dendroceros.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]&lt;br&gt;
Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300

Classes: Anthocerotopsida, Leiosporocerotopsida
===Bryophytes===
[[File:Tionesta-ac-moss2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Bryophyta]]&lt;br&gt;
Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

Classes: Andreaeobryopsida,
Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida

===Chlorophytes===
[[File:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Chlorophyta]]&lt;br&gt;
Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000
Classes: Chlorodendrophyceae, Chlorophyceae, Chloropicophyceae, †Chuariophyceae, Mamiellophyceae, Nephroselmidophyceae, Pedinophyceae, Picocystophyceae, Pyramimonadophyceae, Trebouxiophyceae, Ulvophyceae
===Cycadophytes===
[[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Cycadophyta]]&lt;br&gt;
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

Classes: Cycadopsida

==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>3noajh5bw9wzo4uvx89oe16mm9jt0ah</sha1>
    </revision>
    <revision>
      <id>2834710</id>
      <parentid>2834707</parentid>
      <timestamp>2026-09-27T20:23:43Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Ginkgo added.</comment>
      <origin>2834710</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10957" sha1="8aqm1fwldlwfh27a13mguct6wc2v1o2" xml:space="preserve">[[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px]]
Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.&lt;ref name=Jeffrey1967&gt;{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt;
==Information==
Name Meaning: Twisted plant

English Common Name: Streptophytes

Major distinguishing characteristics: 

Approximate number of species described: &gt;350,000
==Classification==
The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.&lt;ref&gt;{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}&lt;/ref&gt;{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,&lt;ref name=Jeffrey1967/&gt;&lt;ref name=Adletal2012&gt;{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}&lt;/ref&gt; Streptophytina); others include more groups (e.g., Charophyta&lt;!-- Karol et al. 2009--&gt;&lt;ref name=Adletal2012/&gt; Streptophyta,&lt;ref name=Bremer1985&gt;{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}&lt;/ref&gt;&lt;!-- de Reviers 2002; --&gt;&lt;ref name=Leliaert-2012&gt;{{cite journal  |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}&lt;/ref&gt; Streptobionta&lt;ref&gt;{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}&lt;/ref&gt;); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,&lt;ref name=&quot;Cavalier-Smith1993&quot;&gt;{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}&lt;/ref&gt;&lt;ref name=Leliaert-2012/&gt; Charophyceae{{citation needed|reason=Mattox &amp; Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.&lt;ref name=Liang2020&gt;{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}&lt;/ref&gt;&lt;ref name=Nedelcu2006&gt;{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}&lt;/ref&gt; 
==Phylogeny==
Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.&lt;ref&gt;{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology &amp; Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt; (Warning, this phylogenetic picture needs to be improved (See [[Talk:Streptophytes|talk]]).)

{{clade
  |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]`
|2={{clade
      |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]]
|2={{clade
|1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]]
  |2={{clade
    |1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]]
    |label2=Phragmoplastophyta
    |2={{clade
      |1=Charophyceae (Stoneworts &amp; musk grasses) [[File:CharaFragilis.jpg|60px]]
      |2={{clade
          |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]]
        |2={{clade
        |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]]          |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]]
           }}
         }}
       }}
     }}
   }}
}}}}

==Diversity==
There are many species of Streptophytes. These are just the phyla:
===Anthocerotophytes===
[[File:Dendroceros.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]&lt;br&gt;
Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300

Classes: Anthocerotopsida, Leiosporocerotopsida
===Bryophytes===
[[File:Tionesta-ac-moss2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Bryophyta]]&lt;br&gt;
Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

Classes: Andreaeobryopsida,
Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida

===Chlorophytes===
[[File:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Chlorophyta]]&lt;br&gt;
Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000

Classes: Chlorodendrophyceae, Chlorophyceae, Chloropicophyceae, †Chuariophyceae, Mamiellophyceae, Nephroselmidophyceae, Pedinophyceae, Picocystophyceae, Pyramimonadophyceae, Trebouxiophyceae, Ulvophyceae
===Cycadophytes===
[[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Cycadophyta]]&lt;br&gt;
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

Classes: Cycadopsida
===Ginkgophytes===
[[File:Ginkgo biloba in Lucenec2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Ginkgophyta]]&lt;br&gt;
Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

Classes: Ginkgoopsida
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>8aqm1fwldlwfh27a13mguct6wc2v1o2</sha1>
    </revision>
    <revision>
      <id>2834712</id>
      <parentid>2834710</parentid>
      <timestamp>2026-09-27T20:37:38Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Yes! I finally improved the phylogeny!!!!!</comment>
      <origin>2834712</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10904" sha1="2oxzo11inhchdh2w6wjuz7uyzwf358a" xml:space="preserve">[[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px]]
Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.&lt;ref name=Jeffrey1967&gt;{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt;
==Information==
Name Meaning: Twisted plant

English Common Name: Streptophytes

Major distinguishing characteristics: 

Approximate number of species described: &gt;350,000
==Classification==
The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.&lt;ref&gt;{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}&lt;/ref&gt;{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,&lt;ref name=Jeffrey1967/&gt;&lt;ref name=Adletal2012&gt;{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}&lt;/ref&gt; Streptophytina); others include more groups (e.g., Charophyta&lt;!-- Karol et al. 2009--&gt;&lt;ref name=Adletal2012/&gt; Streptophyta,&lt;ref name=Bremer1985&gt;{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}&lt;/ref&gt;&lt;!-- de Reviers 2002; --&gt;&lt;ref name=Leliaert-2012&gt;{{cite journal  |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}&lt;/ref&gt; Streptobionta&lt;ref&gt;{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}&lt;/ref&gt;); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,&lt;ref name=&quot;Cavalier-Smith1993&quot;&gt;{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}&lt;/ref&gt;&lt;ref name=Leliaert-2012/&gt; Charophyceae{{citation needed|reason=Mattox &amp; Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.&lt;ref name=Liang2020&gt;{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}&lt;/ref&gt;&lt;ref name=Nedelcu2006&gt;{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}&lt;/ref&gt; 
==Phylogeny==
Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.&lt;ref&gt;{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology &amp; Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt; 
{{clade
|label1=Streptophyta
|1={{clade
   |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]`
|2={{clade
      |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]]
|2={{clade
|1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]]
  |2={{clade
    
|1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]]
    |label2=Phragmoplastophyta
    |2={{clade
      |1=Charophyceae (Stoneworts &amp; musk grasses) [[File:CharaFragilis.jpg|60px]]
      |2={{clade
          |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]]
        |2={{clade
        |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]]          |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]]
           }}
         }}
       }}
     }}
   }}
   }}
}}}}

==Diversity==
There are many species of Streptophytes. These are just the phyla:
===Anthocerotophytes===
[[File:Dendroceros.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]&lt;br&gt;
Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300

Classes: Anthocerotopsida, Leiosporocerotopsida
===Bryophytes===
[[File:Tionesta-ac-moss2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Bryophyta]]&lt;br&gt;
Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

Classes: Andreaeobryopsida,
Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida

===Chlorophytes===
[[File:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Chlorophyta]]&lt;br&gt;
Name Meaning: Yellow-green plant

English Common Name: Chlorophytes

Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as &quot;higher&quot; plant divisions

Approximate number of species described: 8,000

Classes: Chlorodendrophyceae, Chlorophyceae, Chloropicophyceae, †Chuariophyceae, Mamiellophyceae, Nephroselmidophyceae, Pedinophyceae, Picocystophyceae, Pyramimonadophyceae, Trebouxiophyceae, Ulvophyceae
===Cycadophytes===
[[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Cycadophyta]]&lt;br&gt;
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

Classes: Cycadopsida
===Ginkgophytes===
[[File:Ginkgo biloba in Lucenec2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Ginkgophyta]]&lt;br&gt;
Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

Classes: Ginkgoopsida
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>2oxzo11inhchdh2w6wjuz7uyzwf358a</sha1>
    </revision>
    <revision>
      <id>2834762</id>
      <parentid>2834712</parentid>
      <timestamp>2026-09-28T01:23:45Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Turns out Chlorophytes are not streptophytes. (At least there's Gnetophyta to replace it.)</comment>
      <origin>2834762</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10642" sha1="4p7xqy08k9nf6cjhh82ghjjekearg5h" xml:space="preserve">[[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px]]
Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.&lt;ref name=Jeffrey1967&gt;{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt;
==Information==
Name Meaning: Twisted plant

English Common Name: Streptophytes

Major distinguishing characteristics: 

Approximate number of species described: &gt;350,000
==Classification==
The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.&lt;ref&gt;{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}&lt;/ref&gt;{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,&lt;ref name=Jeffrey1967/&gt;&lt;ref name=Adletal2012&gt;{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}&lt;/ref&gt; Streptophytina); others include more groups (e.g., Charophyta&lt;!-- Karol et al. 2009--&gt;&lt;ref name=Adletal2012/&gt; Streptophyta,&lt;ref name=Bremer1985&gt;{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}&lt;/ref&gt;&lt;!-- de Reviers 2002; --&gt;&lt;ref name=Leliaert-2012&gt;{{cite journal  |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}&lt;/ref&gt; Streptobionta&lt;ref&gt;{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}&lt;/ref&gt;); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,&lt;ref name=&quot;Cavalier-Smith1993&quot;&gt;{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}&lt;/ref&gt;&lt;ref name=Leliaert-2012/&gt; Charophyceae{{citation needed|reason=Mattox &amp; Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.&lt;ref name=Liang2020&gt;{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}&lt;/ref&gt;&lt;ref name=Nedelcu2006&gt;{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}&lt;/ref&gt; 
==Phylogeny==
Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.&lt;ref&gt;{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology &amp; Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt; 
{{clade
|label1=Streptophyta
|1={{clade
   |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]`
|2={{clade
      |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]]
|2={{clade
|1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]]
  |2={{clade
    
|1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]]
    |label2=Phragmoplastophyta
    |2={{clade
      |1=Charophyceae (Stoneworts &amp; musk grasses) [[File:CharaFragilis.jpg|60px]]
      |2={{clade
          |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]]
        |2={{clade
        |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]]          |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]]
           }}
         }}
       }}
     }}
   }}
   }}
}}}}

==Diversity==
There are many species of Streptophytes. These are just the phyla:
===Anthocerotophytes===
[[File:Dendroceros.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]&lt;br&gt;
Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300

Classes: Anthocerotopsida, Leiosporocerotopsida
===Bryophytes===
[[File:Tionesta-ac-moss2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Bryophyta]]&lt;br&gt;
Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

Classes: Andreaeobryopsida,
Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida

===Cycadophytes===
[[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Cycadophyta]]&lt;br&gt;
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

Classes: Cycadopsida
===Ginkgophytes===
[[File:Ginkgo biloba in Lucenec2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Ginkgophyta]]&lt;br&gt;
Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

Classes: Ginkgoopsida
===Gnetophytes===
[[File:Welwitschia at Ugab River basin.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Gnetophyta]]&lt;br&gt;
Name Meaning: Gnetum-like plant

English Common Name: Gnetophytes

Major distinguishing characteristics: Seeds and woody vascular system with vessels.

Approximate number of species described: 70

Classes: Gnetopsida—Actually, it's a subclass: Gnetidae
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>4p7xqy08k9nf6cjhh82ghjjekearg5h</sha1>
    </revision>
    <revision>
      <id>2834777</id>
      <parentid>2834762</parentid>
      <timestamp>2026-09-28T02:02:31Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Great!</comment>
      <origin>2834777</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10688" sha1="6hlwfbpc0ttu8kdhassreroxd37211e" xml:space="preserve">[[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px|Different Plant species from different phyla.]]
Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.&lt;ref name=Jeffrey1967&gt;{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt;
==Information==
Name Meaning: Twisted plant

English Common Name: Streptophytes

Major distinguishing characteristics: 

Approximate number of species described: &gt;350,000
==Classification==
The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.&lt;ref&gt;{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}&lt;/ref&gt;{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,&lt;ref name=Jeffrey1967/&gt;&lt;ref name=Adletal2012&gt;{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}&lt;/ref&gt; Streptophytina); others include more groups (e.g., Charophyta&lt;!-- Karol et al. 2009--&gt;&lt;ref name=Adletal2012/&gt; Streptophyta,&lt;ref name=Bremer1985&gt;{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}&lt;/ref&gt;&lt;!-- de Reviers 2002; --&gt;&lt;ref name=Leliaert-2012&gt;{{cite journal  |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}&lt;/ref&gt; Streptobionta&lt;ref&gt;{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}&lt;/ref&gt;); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,&lt;ref name=&quot;Cavalier-Smith1993&quot;&gt;{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}&lt;/ref&gt;&lt;ref name=Leliaert-2012/&gt; Charophyceae{{citation needed|reason=Mattox &amp; Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.&lt;ref name=Liang2020&gt;{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}&lt;/ref&gt;&lt;ref name=Nedelcu2006&gt;{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}&lt;/ref&gt; 
==Phylogeny==
Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.&lt;ref&gt;{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology &amp; Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}&lt;/ref&gt; 
{{clade
|label1=Streptophyta
|1={{clade
   |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]`
|2={{clade
      |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]]
|2={{clade
|1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]]
  |2={{clade
    
|1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]]
    |label2=Phragmoplastophyta
    |2={{clade
      |1=Charophyceae (Stoneworts &amp; musk grasses) [[File:CharaFragilis.jpg|60px]]
      |2={{clade
          |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]]
        |2={{clade
        |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]]          |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]]
           }}
         }}
       }}
     }}
   }}
   }}
}}}}

==Diversity==
There are many species of Streptophytes. These are just the phyla:
===Anthocerotophytes===
[[File:Dendroceros.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]&lt;br&gt;
Name Meaning: Anthoceros-like plant

English Common Name: Hornworts

Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system

Approximate number of species described: 100-300

Classes: Anthocerotopsida, Leiosporocerotopsida
===Bryophytes===
[[File:Tionesta-ac-moss2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Bryophyta]]&lt;br&gt;
Name Meaning: Bryum-like plant, moss plant

English Common Name: Moss

Major distinguishing characteristics: Persistent branched sporophytes, no vascular system

Approximate number of species described: 12,000

Classes: Andreaeobryopsida,
Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida

===Cycadophytes===
[[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Cycadophyta]]&lt;br&gt;
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200

Classes: Cycadopsida
===Ginkgophytes===
[[File:Ginkgo biloba in Lucenec2.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Ginkgophyta]]&lt;br&gt;
Name Meaning: Ginkgo-like plant

English Common Name: Ginkgo, maidenhair tree

Major distinguishing characteristics: Seeds not protected by fruit

Approximate number of species described: 1 living, about 50 extinct

Classes: Ginkgoopsida
===Gnetophytes===
[[File:Welwitschia at Ugab River basin.jpg|thumb|200x200px]]
Main article: [[Plant Divisions (Phyla)/Gnetophyta]]&lt;br&gt;
Name Meaning: Gnetum-like plant

English Common Name: Gnetophytes

Major distinguishing characteristics: Seeds and woody vascular system with vessels.

Approximate number of species described: 70

Classes: Gnetopsida—Actually, it's a subclass: Gnetidae
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>6hlwfbpc0ttu8kdhassreroxd37211e</sha1>
    </revision>
  </page>
  <page>
    <title>The Lost Zodiac</title>
    <ns>0</ns>
    <id>332225</id>
    <revision>
      <id>2834682</id>
      <parentid>2834475</parentid>
      <timestamp>2026-09-27T15:57:01Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <minor />
      <comment>202 Xup iartkab lakarg moved page [[User:202 Xup iartkab lakarg]] to [[The Lost Zodiac]]</comment>
      <origin>2834475</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9184" sha1="d0rkqk2rn5cnukfljrswlxq77j5ml91" xml:space="preserve">{{Infobox
| title = The Lost Zodiac
| image =
| caption =

| label1 = Author
| data1 = Tahmid Arian Borno

| label2 = Year
| data2 = 2026

| label3 = Status
| data3 = Proposed hypothesis; under development

| label4 = Subject
| data4 = Astrology / stellar symbolism
}}

== The Lost Zodiac ==
'''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026.
It is an experimental astrological framework based on selected stars and
constellational figures, their associated date ranges, and a proposed method
for converting conventional natal-chart positions into Lost Zodiac positions.

&lt;div style=&quot;clear: both; width: auto; margin: 1em 0;&quot;&gt;

&lt;!-- Research Project Box --&gt;
&lt;div style=&quot;border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This is a research project at Wikiversity.'''
&lt;/div&gt;

&lt;!-- Hypothesis Warning Box --&gt;
&lt;div style=&quot;border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.'''
&lt;/div&gt;

&lt;/div&gt;

== Background ==

The Lost Zodiac proposes a set of zodiacal periods associated with stars and
constellational figures. The system contains 31 periods, built from 22 distinct
constellational figures, several of which recur across non-adjacent periods,
distributed throughout the year.

The repetition of certain constellational figures is intentional. In
InteractiveStars.com's Lost Zodiac, the periods track the Sun's proximity to
individual prominent stars rather than dividing the sky into continuous,
side-by-side sectors like the traditional zodiac. Consequently, the same
constellation may appear in multiple non-adjacent periods when different stars
associated with it are involved.

The system was created by astrologer Catherine Tennant and features 22
&quot;lost&quot; star signs based on major constellations located north and south of the
main zodiac belt. Source: InteractiveStars.com, &quot;The Lost Zodiac&quot; (accessed
September 23, 2026). Link to be added once this account is autoconfirmed.

Period boundaries, associated constellational figures, and star assignments
are sourced from InteractiveStars.com. The explanation of the recurring
constellational figures is also based on the design of that source system.
The proposed organization of the periods within this article and the
natal-chart conversion methodology below are original to the author.
Per-sign source links for each row of the table below will be added once
this account reaches autoconfirmed status.

== The Lost Zodiac Signs ==

{| class=&quot;wikitable sortable&quot;
! Sign
! Period
! Associated star(s)
! Source
|-
| The Lyre of Orpheus
| December 29 – January 13
| Vega
| InteractiveStars.com
|-
| The Eagle
| January 14 – January 28
| Altair
| InteractiveStars.com
|-
| The Dolphin
| January 29 – February 8
| Rotanev and Sualocin
| InteractiveStars.com
|-
| The Swan
| February 9 – February 28/29
| Sadir
| InteractiveStars.com
|-
| The River of Night
| March 1 – March 12
| Achernar
| InteractiveStars.com
|-
| Pegasus
| March 13 – April 1
| Markab, Scheat and Algenib
| InteractiveStars.com
|-
| Andromeda
| April 2 – April 9
| Alpheratz
| InteractiveStars.com
|-
| The River of Night
| April 10 – April 18
| Acamar
| InteractiveStars.com
|-
| Andromeda
| April 19 – May 8
| Mirach and Alamach
| InteractiveStars.com
|-
| The River of Night
| May 9 – May 15
| Rana and Zanrak
| InteractiveStars.com
|-
| Perseus
| May 16 – May 31
| Algol and Mirfak
| InteractiveStars.com
|-
| Orion
| June 1 – June 7
| Rigel
| InteractiveStars.com
|-
| The Charioteer
| June 8 – June 16
| Capella
| InteractiveStars.com
|-
| Orion
| June 17 – June 27
| Betelgeuse
| InteractiveStars.com
|-
| The Dogs
| June 28 – July 7
| Sirius
| InteractiveStars.com
|-
| The Ship of the Argonauts
| July 8 – July 17
| Canopus
| InteractiveStars.com
|-
| The Dogs
| July 18 – July 25
| Procyon
| InteractiveStars.com
|-
| The Dragon
| July 26 – August 7
| Gianfar
| InteractiveStars.com
|-
| The Great Bear
| August 8 – August 15
| Dubhe and Merak
| InteractiveStars.com
|-
| The Sea Serpent
| August 16 – August 23
| Alphard
| InteractiveStars.com
|-
| The Great Bear
| August 24 – September 10
| Phekda, Megrez, Alioth and Mizar
| InteractiveStars.com
|-
| The Cup
| September 11 – September 21
| Alkes
| InteractiveStars.com
|-
| The Ship of the Argonauts
| September 22 – September 28
| Markeb
| InteractiveStars.com
|-
| The Raven
| September 29 – October 11
| Minkar and Algorab
| InteractiveStars.com
|-
| The Bear Keeper
| October 12 – October 26
| Arcturus and Izar
| InteractiveStars.com
|-
| The Crown of the North Wind
| October 27 – November 10
| Alphecca
| InteractiveStars.com
|-
| The Serpent
| November 11 – November 19
| Unuk Elhaia
| InteractiveStars.com
|-
| The Wise Centaur
| November 20 – December 5
| Toliman (Alpha Centauri)
| InteractiveStars.com
|-
| Ophiuchus
| December 6 – December 16
| Han, Sabik and Ras Alhague
| InteractiveStars.com
|-
| The Dragon
| December 17 – December 23
| Grumium and Etanin
| InteractiveStars.com
|-
| The Serpent
| December 24 – December 28
| Alya
| InteractiveStars.com
|}

== Natal-chart conversion methodology ==

The Lost Zodiac proposes a proportional conversion between a conventional
zodiacal position and the corresponding Lost Zodiac period.

A placement is first treated as a position within its conventional zodiac sign.
That position is converted into a corresponding calendar date and time. The
resulting date and time are then located within the relevant Lost Zodiac period.

Each listed period includes the full span of every calendar date in its range.
The starting date begins at 12:00 a.m., and the ending date concludes at
11:59 p.m. For example, a period ending on February 28 includes the entire
day of February 28. In leap years, the Swan period ends at 11:59 p.m. on
February 29.

For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and
the end of the period as 29°59′. The beginning boundary is midnight at the
start of the first listed date, while the ending boundary is 11:59 p.m. on
the last listed date.

The proposed formula is:

&lt;math&gt;
L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59'
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; = resulting Lost Zodiac degree
* &lt;math&gt;t&lt;/math&gt; = the corresponding date and time of the natal placement
* &lt;math&gt;t_0&lt;/math&gt; = beginning of the Lost Zodiac period
* &lt;math&gt;t_1&lt;/math&gt; = end boundary of the Lost Zodiac period

=== Example ===

A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009
at approximately 6:25 AM in the example used by the author.

The Swan period runs from February 9 through the last day of February,
ending on February 28 in a common year or February 29 in a leap year.

The position of February 20 at 6:25 AM is approximately 56.34% through the
Swan period.

Therefore:

&lt;math&gt;
29^\circ 59'\times0.5634\approx16^\circ 53'
&lt;/math&gt;

The resulting proposed Lost Zodiac position is therefore:

'''Swan 16°53′'''

== Reference natal chart ==

The following conventional natal-chart placements are used as the reference
positions for the Lost Zodiac conversion.

{| class=&quot;wikitable&quot;
! Natal point
! Conventional natal position
! House
|-
| Sun
| Pisces 1°29′
| 1st House
|-
| Ascendant
| Aquarius 29°24′
| —
|-
| Moon
| Capricorn 5°24′
| 11th House
|-
| Mercury
| Aquarius 6°07′
| 12th House
|-
| Venus
| Aries 11°30′
| 2nd House
|-
| Mars
| Aquarius 11°55′
| 12th House
|-
| Jupiter
| Aquarius 10°38′
| 12th House
|-
| Saturn
| Virgo 19°41′ Rx
| 7th House
|-
| Uranus
| Pisces 21°25′
| 1st House
|-
| Neptune
| Aquarius 24°13′
| 12th House
|-
| Pluto
| Capricorn 2°47′
| 10th House
|-
| North Node
| Aquarius 8°18′ Rx
| 12th House
|-
| Lilith
| Capricorn 5°16′
| 11th House
|-
| Chiron
| Aquarius 21°52′
| 12th House
|-
| Fortune
| Aries 25°29′
| 2nd House
|-
| Vertex
| Virgo 17°15′
| 7th House
|-
| MC
| Sagittarius 8°28′
| —
|}

== Example: author's Lost Zodiac natal chart ==

Using the conversion methodology described above, the author's proposed
Lost Zodiac natal placements are:

{| class=&quot;wikitable&quot;
! Natal point
! Lost Zodiac placement
|-
| Sun
| Swan 16°53′
|-
| Ascendant
| Swan 14°06′
|-
| Moon
| Serpent 20°23′
|-
| Mercury
| Eagle 24°13′
|-
| Venus
| Pegasus 29°14′
|-
| Mars
| Dolphin 7°57′
|-
| Jupiter
| Dolphin 4°27′
|-
| Saturn
| Cup 1°52′
|-
| Uranus
| River of Night 28°32′
|-
| Neptune
| Swan 6°19′
|-
| Pluto
| Serpent 4°42′
|-
| North Node
| Eagle 28°35′
|-
| Lilith
| Serpent 19°35′
|-
| Chiron
| Swan 2°48′
|-
| Fortune
| River of Night 18°16′
|-
| Vertex
| Great Bear 27°04′
|-
| MC
| Wise Centaur 19°37′
|}</text>
      <sha1>d0rkqk2rn5cnukfljrswlxq77j5ml91</sha1>
    </revision>
    <revision>
      <id>2834687</id>
      <parentid>2834682</parentid>
      <timestamp>2026-09-27T16:59:56Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <origin>2834687</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13292" sha1="9d9fpli6zakjuut5eefpxqut22e5oaq" xml:space="preserve">{{Infobox
| title = The Lost Zodiac
| image =
| caption =

| label1 = Author
| data1 = Tahmid Arian Borno

| label2 = Year
| data2 = 2026

| label3 = Status
| data3 = Proposed hypothesis; under development

| label4 = Subject
| data4 = Astrology / stellar symbolism
}}

== The Lost Zodiac ==
'''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026.
It is an experimental astrological framework based on selected stars and
constellational figures, their associated date ranges, and a proposed method
for converting conventional natal-chart positions into Lost Zodiac positions.

&lt;div style=&quot;clear: both; width: auto; margin: 1em 0;&quot;&gt;

&lt;!-- Research Project Box --&gt;
&lt;div style=&quot;border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This is a research project at Wikiversity.'''
&lt;/div&gt;

&lt;!-- Hypothesis Warning Box --&gt;
&lt;div style=&quot;border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.'''
&lt;/div&gt;

&lt;/div&gt;

== Background ==

The Lost Zodiac proposes a set of zodiacal periods associated with stars and
constellational figures. The system contains 31 periods, built from 22 distinct
constellational figures, several of which recur across non-adjacent periods,
distributed throughout the year.

The repetition of certain constellational figures is intentional. In
InteractiveStars.com's Lost Zodiac, the periods track the Sun's proximity to
individual prominent stars rather than dividing the sky into continuous,
side-by-side sectors like the traditional zodiac. Consequently, the same
constellation may appear in multiple non-adjacent periods when different stars
associated with it are involved.

The system was created by astrologer Catherine Tennant and features 22
&quot;lost&quot; star signs based on major constellations located north and south of the
main zodiac belt. Source: InteractiveStars.com, &quot;The Lost Zodiac&quot; (accessed
September 23, 2026). Link to be added once this account is autoconfirmed.

Period boundaries, associated constellational figures, and star assignments
are sourced from InteractiveStars.com. The explanation of the recurring
constellational figures is also based on the design of that source system.
The proposed organization of the periods within this article and the
natal-chart conversion methodology below are original to the author.
Per-sign source links for each row of the table below will be added once
this account reaches autoconfirmed status.

== The Lost Zodiac Signs ==

{| class=&quot;wikitable sortable&quot;
! Sign
! Period
! Associated star(s)
! Sources
|-
| The Lyre of Orpheus
| December 29 – January 13
| Vega
| &lt;ref name=&quot;The Lyre of Orpheus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&amp;date=Dec-29th@Jan-13th
 |title=The Lyre of Orpheus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Eagle
| January 14 – January 28
| Altair
| &lt;ref name=&quot;The Eagle&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf
 |title=The Eagle
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dolphin
| January 29 – February 8
| Rotanev and Sualocin
| &lt;ref name=&quot;The Dolphin&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&amp;date=Jan-29th@Feb-8th
 |title=The Dolphin
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Swan
| February 9 – February 28/29
| Sadir
| &lt;ref name=&quot;The Swan&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf
 |title=The Swan
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| March 1 – March 12
| Achernar
| &lt;ref name=&quot;The River of Night&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf
 |title=The River of Night
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Pegasus
| March 13 – April 1
| Markab, Scheat and Algenib
| &lt;ref name=&quot;Pegasus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf
 |title=Pegasus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Andromeda
| April 2 – April 9
| Alpheratz
| &lt;ref name=&quot;Andromeda&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf
 |title=Andromeda
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| April 10 – April 18
| Acamar
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Andromeda
| April 19 – May 8
| Mirach and Alamach
| &lt;ref name=&quot;Andromeda&quot;/&gt;
|-
| The River of Night
| May 9 – May 15
| Rana and Zanrak
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Perseus
| May 16 – May 31
| Algol and Mirfak
| &lt;ref name=&quot;Perseus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&amp;date=May16th@31st
 |title=Perseus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 1 – June 7
| Rigel
| &lt;ref name=&quot;Orion&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf
 |title=Orion
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Charioteer
| June 8 – June 16
| Capella
| &lt;ref name=&quot;The Charioteer&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf
 |title=The Charioteer
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 17 – June 27
| Betelgeuse
| &lt;ref name=&quot;Orion&quot;/&gt;
|-
| The Dogs
| June 28 – July 7
| Sirius
| &lt;ref name=&quot;The Dogs&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf
 |title=The Dogs
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| July 8 – July 17
| Canopus
| &lt;ref name=&quot;Ship of the Argonauts&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf
 |title=The Ship of the Argonauts
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dogs
| July 18 – July 25
| Procyon
| &lt;ref name=&quot;The Dogs&quot;/&gt;
|-
| The Dragon
| July 26 – August 7
| Gianfar
| &lt;ref name=&quot;The Dragon&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 8 – August 15
| Dubhe and Merak
| &lt;ref name=&quot;The Great Bear&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&amp;date=Aug-8th@15th
 |title=The Great Bear
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Sea Serpent
| August 16 – August 23
| Alphard
| &lt;ref name=&quot;The Sea Serpent&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf
 |title=The Sea Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 24 – September 10
| Phekda, Megrez, Alioth and Mizar
| &lt;ref name=&quot;The Great Bear&quot;/&gt;
|-
| The Cup
| September 11 – September 21
| Alkes
| &lt;ref name=&quot;The Cup&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf
 |title=The Cup
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| September 22 – September 28
| Markeb
| &lt;ref name=&quot;Ship of the Argonauts&quot;/&gt;
|-
| The Raven
| September 29 – October 11
| Minkar and Algorab
| &lt;ref name=&quot;The Raven&quot;&gt;{{cite web
 |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72
 |title=The Raven
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Bear Keeper
| October 12 – October 26
| Arcturus and Izar
| &lt;ref name=&quot;The Bear Keeper&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf
 |title=The Bear Keeper
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Crown of the North Wind
| October 27 – November 10
| Alphecca
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf
 |title=The Lost Zodiac
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| November 11 – November 19
| Unuk Elhaia
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Wise Centaur
| November 20 – December 5
| Toliman (Alpha Centauri)
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf
 |title=The Wise Centaur
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Ophiuchus
| December 6 – December 16
| Han, Sabik and Ras Alhague
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf
 |title=Ophiuchus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dragon
| December 17 – December 23
| Grumium and Etanin
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| December 24 – December 28
| Alya
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|}

== Natal-chart conversion methodology ==

The Lost Zodiac proposes a proportional conversion between a conventional
zodiacal position and the corresponding Lost Zodiac period.

A placement is first treated as a position within its conventional zodiac sign.
That position is converted into a corresponding calendar date and time. The
resulting date and time are then located within the relevant Lost Zodiac period.

Each listed period includes the full span of every calendar date in its range.
The starting date begins at 12:00 a.m., and the ending date concludes at
11:59 p.m. For example, a period ending on February 28 includes the entire
day of February 28. In leap years, the Swan period ends at 11:59 p.m. on
February 29.

For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and
the end of the period as 29°59′. The beginning boundary is midnight at the
start of the first listed date, while the ending boundary is 11:59 p.m. on
the last listed date.

The proposed formula is:

&lt;math&gt;
L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59'
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; = resulting Lost Zodiac degree
* &lt;math&gt;t&lt;/math&gt; = the corresponding date and time of the natal placement
* &lt;math&gt;t_0&lt;/math&gt; = beginning of the Lost Zodiac period
* &lt;math&gt;t_1&lt;/math&gt; = end boundary of the Lost Zodiac period

=== Example ===

A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009
at approximately 6:25 AM in the example used by the author.

The Swan period runs from February 9 through the last day of February,
ending on February 28 in a common year or February 29 in a leap year.

The position of February 20 at 6:25 AM is approximately 56.34% through the
Swan period.

Therefore:

&lt;math&gt;
29^\circ 59'\times0.5634\approx16^\circ 53'
&lt;/math&gt;

The resulting proposed Lost Zodiac position is therefore:

'''Swan 16°53′'''

== Reference natal chart ==

The following conventional natal-chart placements are used as the reference
positions for the Lost Zodiac conversion.

{| class=&quot;wikitable&quot;
! Natal point
! Conventional natal position
! House
|-
| Sun
| Pisces 1°29′
| 1st House
|-
| Ascendant
| Aquarius 29°24′
| —
|-
| Moon
| Capricorn 5°24′
| 11th House
|-
| Mercury
| Aquarius 6°07′
| 12th House
|-
| Venus
| Aries 11°30′
| 2nd House
|-
| Mars
| Aquarius 11°55′
| 12th House
|-
| Jupiter
| Aquarius 10°38′
| 12th House
|-
| Saturn
| Virgo 19°41′ Rx
| 7th House
|-
| Uranus
| Pisces 21°25′
| 1st House
|-
| Neptune
| Aquarius 24°13′
| 12th House
|-
| Pluto
| Capricorn 2°47′
| 10th House
|-
| North Node
| Aquarius 8°18′ Rx
| 12th House
|-
| Lilith
| Capricorn 5°16′
| 11th House
|-
| Chiron
| Aquarius 21°52′
| 12th House
|-
| Fortune
| Aries 25°29′
| 2nd House
|-
| Vertex
| Virgo 17°15′
| 7th House
|-
| MC
| Sagittarius 8°28′
| —
|}

== Example: author's Lost Zodiac natal chart ==

Using the conversion methodology described above, the author's proposed
Lost Zodiac natal placements are:

{| class=&quot;wikitable&quot;
! Natal point
! Lost Zodiac placement
|-
| Sun
| Swan 16°53′
|-
| Ascendant
| Swan 14°06′
|-
| Moon
| Serpent 20°23′
|-
| Mercury
| Eagle 24°13′
|-
| Venus
| Pegasus 29°14′
|-
| Mars
| Dolphin 7°57′
|-
| Jupiter
| Dolphin 4°27′
|-
| Saturn
| Cup 1°52′
|-
| Uranus
| River of Night 28°32′
|-
| Neptune
| Swan 6°19′
|-
| Pluto
| Serpent 4°42′
|-
| North Node
| Eagle 28°35′
|-
| Lilith
| Serpent 19°35′
|-
| Chiron
| Swan 2°48′
|-
| Fortune
| River of Night 18°16′
|-
| Vertex
| Great Bear 27°04′
|-
| MC
| Wise Centaur 19°37′
|}</text>
      <sha1>9d9fpli6zakjuut5eefpxqut22e5oaq</sha1>
    </revision>
    <revision>
      <id>2834689</id>
      <parentid>2834687</parentid>
      <timestamp>2026-09-27T17:12:49Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <comment>/* Background */</comment>
      <origin>2834689</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13037" sha1="5vceay6mfau7iw43htolfocob7q5zry" xml:space="preserve">{{Infobox
| title = The Lost Zodiac
| image =
| caption =

| label1 = Author
| data1 = Tahmid Arian Borno

| label2 = Year
| data2 = 2026

| label3 = Status
| data3 = Proposed hypothesis; under development

| label4 = Subject
| data4 = Astrology / stellar symbolism
}}

== The Lost Zodiac ==
'''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026.
It is an experimental astrological framework based on selected stars and
constellational figures, their associated date ranges, and a proposed method
for converting conventional natal-chart positions into Lost Zodiac positions.

&lt;div style=&quot;clear: both; width: auto; margin: 1em 0;&quot;&gt;

&lt;!-- Research Project Box --&gt;
&lt;div style=&quot;border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This is a research project at Wikiversity.'''
&lt;/div&gt;

&lt;!-- Hypothesis Warning Box --&gt;
&lt;div style=&quot;border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.'''
&lt;/div&gt;

&lt;/div&gt;

== Background ==

The Lost Zodiac is based on a set of zodiacal periods associated with prominent stars and constellational figures. The source system on which the reconstruction is based contains 31 periods representing 22 distinct constellational figures. Several figures recur in non-adjacent periods because different stars associated with the same figure are assigned to different parts of the year.

The period boundaries, associated figures, and principal stars used in this project are derived from the Lost Zodiac material published by InteractiveStars.com.&lt;ref name=&quot;InteractiveStars&quot;&gt;{{cite web
|url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php
|title=The Lost Zodiac
|publisher=InteractiveStars.com
|access-date=2026-09-23
}}&lt;/ref&gt;

The source material describes the system as a set of &quot;lost&quot; star signs based on major constellations located north and south of the main zodiac belt.&lt;ref name=&quot;InteractiveStars&quot;/&gt;

The present project does not claim that the organization of these periods or their relationship to the modern natal zodiac has been historically established. Instead, the source material is used as the astronomical and symbolic foundation for a proposed reconstruction.

== The Lost Zodiac Signs ==

{| class=&quot;wikitable sortable&quot;
! Sign
! Period
! Associated star(s)
! Sources
|-
| The Lyre of Orpheus
| December 29 – January 13
| Vega
| &lt;ref name=&quot;The Lyre of Orpheus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&amp;date=Dec-29th@Jan-13th
 |title=The Lyre of Orpheus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Eagle
| January 14 – January 28
| Altair
| &lt;ref name=&quot;The Eagle&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf
 |title=The Eagle
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dolphin
| January 29 – February 8
| Rotanev and Sualocin
| &lt;ref name=&quot;The Dolphin&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&amp;date=Jan-29th@Feb-8th
 |title=The Dolphin
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Swan
| February 9 – February 28/29
| Sadir
| &lt;ref name=&quot;The Swan&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf
 |title=The Swan
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| March 1 – March 12
| Achernar
| &lt;ref name=&quot;The River of Night&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf
 |title=The River of Night
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Pegasus
| March 13 – April 1
| Markab, Scheat and Algenib
| &lt;ref name=&quot;Pegasus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf
 |title=Pegasus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Andromeda
| April 2 – April 9
| Alpheratz
| &lt;ref name=&quot;Andromeda&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf
 |title=Andromeda
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| April 10 – April 18
| Acamar
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Andromeda
| April 19 – May 8
| Mirach and Alamach
| &lt;ref name=&quot;Andromeda&quot;/&gt;
|-
| The River of Night
| May 9 – May 15
| Rana and Zanrak
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Perseus
| May 16 – May 31
| Algol and Mirfak
| &lt;ref name=&quot;Perseus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&amp;date=May16th@31st
 |title=Perseus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 1 – June 7
| Rigel
| &lt;ref name=&quot;Orion&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf
 |title=Orion
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Charioteer
| June 8 – June 16
| Capella
| &lt;ref name=&quot;The Charioteer&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf
 |title=The Charioteer
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 17 – June 27
| Betelgeuse
| &lt;ref name=&quot;Orion&quot;/&gt;
|-
| The Dogs
| June 28 – July 7
| Sirius
| &lt;ref name=&quot;The Dogs&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf
 |title=The Dogs
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| July 8 – July 17
| Canopus
| &lt;ref name=&quot;Ship of the Argonauts&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf
 |title=The Ship of the Argonauts
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dogs
| July 18 – July 25
| Procyon
| &lt;ref name=&quot;The Dogs&quot;/&gt;
|-
| The Dragon
| July 26 – August 7
| Gianfar
| &lt;ref name=&quot;The Dragon&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 8 – August 15
| Dubhe and Merak
| &lt;ref name=&quot;The Great Bear&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&amp;date=Aug-8th@15th
 |title=The Great Bear
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Sea Serpent
| August 16 – August 23
| Alphard
| &lt;ref name=&quot;The Sea Serpent&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf
 |title=The Sea Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 24 – September 10
| Phekda, Megrez, Alioth and Mizar
| &lt;ref name=&quot;The Great Bear&quot;/&gt;
|-
| The Cup
| September 11 – September 21
| Alkes
| &lt;ref name=&quot;The Cup&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf
 |title=The Cup
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| September 22 – September 28
| Markeb
| &lt;ref name=&quot;Ship of the Argonauts&quot;/&gt;
|-
| The Raven
| September 29 – October 11
| Minkar and Algorab
| &lt;ref name=&quot;The Raven&quot;&gt;{{cite web
 |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72
 |title=The Raven
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Bear Keeper
| October 12 – October 26
| Arcturus and Izar
| &lt;ref name=&quot;The Bear Keeper&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf
 |title=The Bear Keeper
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Crown of the North Wind
| October 27 – November 10
| Alphecca
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf
 |title=The Lost Zodiac
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| November 11 – November 19
| Unuk Elhaia
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Wise Centaur
| November 20 – December 5
| Toliman (Alpha Centauri)
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf
 |title=The Wise Centaur
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Ophiuchus
| December 6 – December 16
| Han, Sabik and Ras Alhague
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf
 |title=Ophiuchus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dragon
| December 17 – December 23
| Grumium and Etanin
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| December 24 – December 28
| Alya
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|}

== Natal-chart conversion methodology ==

The Lost Zodiac proposes a proportional conversion between a conventional
zodiacal position and the corresponding Lost Zodiac period.

A placement is first treated as a position within its conventional zodiac sign.
That position is converted into a corresponding calendar date and time. The
resulting date and time are then located within the relevant Lost Zodiac period.

Each listed period includes the full span of every calendar date in its range.
The starting date begins at 12:00 a.m., and the ending date concludes at
11:59 p.m. For example, a period ending on February 28 includes the entire
day of February 28. In leap years, the Swan period ends at 11:59 p.m. on
February 29.

For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and
the end of the period as 29°59′. The beginning boundary is midnight at the
start of the first listed date, while the ending boundary is 11:59 p.m. on
the last listed date.

The proposed formula is:

&lt;math&gt;
L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59'
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; = resulting Lost Zodiac degree
* &lt;math&gt;t&lt;/math&gt; = the corresponding date and time of the natal placement
* &lt;math&gt;t_0&lt;/math&gt; = beginning of the Lost Zodiac period
* &lt;math&gt;t_1&lt;/math&gt; = end boundary of the Lost Zodiac period

=== Example ===

A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009
at approximately 6:25 AM in the example used by the author.

The Swan period runs from February 9 through the last day of February,
ending on February 28 in a common year or February 29 in a leap year.

The position of February 20 at 6:25 AM is approximately 56.34% through the
Swan period.

Therefore:

&lt;math&gt;
29^\circ 59'\times0.5634\approx16^\circ 53'
&lt;/math&gt;

The resulting proposed Lost Zodiac position is therefore:

'''Swan 16°53′'''

== Reference natal chart ==

The following conventional natal-chart placements are used as the reference
positions for the Lost Zodiac conversion.

{| class=&quot;wikitable&quot;
! Natal point
! Conventional natal position
! House
|-
| Sun
| Pisces 1°29′
| 1st House
|-
| Ascendant
| Aquarius 29°24′
| —
|-
| Moon
| Capricorn 5°24′
| 11th House
|-
| Mercury
| Aquarius 6°07′
| 12th House
|-
| Venus
| Aries 11°30′
| 2nd House
|-
| Mars
| Aquarius 11°55′
| 12th House
|-
| Jupiter
| Aquarius 10°38′
| 12th House
|-
| Saturn
| Virgo 19°41′ Rx
| 7th House
|-
| Uranus
| Pisces 21°25′
| 1st House
|-
| Neptune
| Aquarius 24°13′
| 12th House
|-
| Pluto
| Capricorn 2°47′
| 10th House
|-
| North Node
| Aquarius 8°18′ Rx
| 12th House
|-
| Lilith
| Capricorn 5°16′
| 11th House
|-
| Chiron
| Aquarius 21°52′
| 12th House
|-
| Fortune
| Aries 25°29′
| 2nd House
|-
| Vertex
| Virgo 17°15′
| 7th House
|-
| MC
| Sagittarius 8°28′
| —
|}

== Example: author's Lost Zodiac natal chart ==

Using the conversion methodology described above, the author's proposed
Lost Zodiac natal placements are:

{| class=&quot;wikitable&quot;
! Natal point
! Lost Zodiac placement
|-
| Sun
| Swan 16°53′
|-
| Ascendant
| Swan 14°06′
|-
| Moon
| Serpent 20°23′
|-
| Mercury
| Eagle 24°13′
|-
| Venus
| Pegasus 29°14′
|-
| Mars
| Dolphin 7°57′
|-
| Jupiter
| Dolphin 4°27′
|-
| Saturn
| Cup 1°52′
|-
| Uranus
| River of Night 28°32′
|-
| Neptune
| Swan 6°19′
|-
| Pluto
| Serpent 4°42′
|-
| North Node
| Eagle 28°35′
|-
| Lilith
| Serpent 19°35′
|-
| Chiron
| Swan 2°48′
|-
| Fortune
| River of Night 18°16′
|-
| Vertex
| Great Bear 27°04′
|-
| MC
| Wise Centaur 19°37′
|}</text>
      <sha1>5vceay6mfau7iw43htolfocob7q5zry</sha1>
    </revision>
    <revision>
      <id>2834690</id>
      <parentid>2834689</parentid>
      <timestamp>2026-09-27T17:18:16Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <comment>/* Background */</comment>
      <origin>2834690</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13042" sha1="39ndctsanku49cl6574t89kvnkbpyub" xml:space="preserve">{{Infobox
| title = The Lost Zodiac
| image =
| caption =

| label1 = Author
| data1 = Tahmid Arian Borno

| label2 = Year
| data2 = 2026

| label3 = Status
| data3 = Proposed hypothesis; under development

| label4 = Subject
| data4 = Astrology / stellar symbolism
}}

== The Lost Zodiac ==
'''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026.
It is an experimental astrological framework based on selected stars and
constellational figures, their associated date ranges, and a proposed method
for converting conventional natal-chart positions into Lost Zodiac positions.

&lt;div style=&quot;clear: both; width: auto; margin: 1em 0;&quot;&gt;

&lt;!-- Research Project Box --&gt;
&lt;div style=&quot;border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This is a research project at Wikiversity.'''
&lt;/div&gt;

&lt;!-- Hypothesis Warning Box --&gt;
&lt;div style=&quot;border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.'''
&lt;/div&gt;

&lt;/div&gt;

== Background ==

The Lost Zodiac is based on a set of zodiacal periods associated with prominent stars and constellational figures. The source system on which the reconstruction is based contains 31 periods representing 22 distinct constellational figures. Several figures recur in non-adjacent periods because different stars associated with the same figure are assigned to different parts of the year.

The period boundaries, associated figures, and principal stars used in this project are derived from the Lost Zodiac material published by InteractiveStars.com.&lt;ref name=&quot;InteractiveStars&quot;&gt;{{cite web
|url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac_sign.php
|title=The Lost Zodiac
|publisher=InteractiveStars.com
|access-date=2026-09-23
}}&lt;/ref&gt;

The source material describes the system as a set of &quot;lost&quot; star signs based on major constellations located north and south of the main zodiac belt.&lt;ref name=&quot;InteractiveStars&quot;/&gt;

The present project does not claim that the organization of these periods or their relationship to the modern natal zodiac has been historically established. Instead, the source material is used as the astronomical and symbolic foundation for a proposed reconstruction.

== The Lost Zodiac Signs ==

{| class=&quot;wikitable sortable&quot;
! Sign
! Period
! Associated star(s)
! Sources
|-
| The Lyre of Orpheus
| December 29 – January 13
| Vega
| &lt;ref name=&quot;The Lyre of Orpheus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&amp;date=Dec-29th@Jan-13th
 |title=The Lyre of Orpheus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Eagle
| January 14 – January 28
| Altair
| &lt;ref name=&quot;The Eagle&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf
 |title=The Eagle
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dolphin
| January 29 – February 8
| Rotanev and Sualocin
| &lt;ref name=&quot;The Dolphin&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&amp;date=Jan-29th@Feb-8th
 |title=The Dolphin
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Swan
| February 9 – February 28/29
| Sadir
| &lt;ref name=&quot;The Swan&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf
 |title=The Swan
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| March 1 – March 12
| Achernar
| &lt;ref name=&quot;The River of Night&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf
 |title=The River of Night
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Pegasus
| March 13 – April 1
| Markab, Scheat and Algenib
| &lt;ref name=&quot;Pegasus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf
 |title=Pegasus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Andromeda
| April 2 – April 9
| Alpheratz
| &lt;ref name=&quot;Andromeda&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf
 |title=Andromeda
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| April 10 – April 18
| Acamar
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Andromeda
| April 19 – May 8
| Mirach and Alamach
| &lt;ref name=&quot;Andromeda&quot;/&gt;
|-
| The River of Night
| May 9 – May 15
| Rana and Zanrak
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Perseus
| May 16 – May 31
| Algol and Mirfak
| &lt;ref name=&quot;Perseus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&amp;date=May16th@31st
 |title=Perseus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 1 – June 7
| Rigel
| &lt;ref name=&quot;Orion&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf
 |title=Orion
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Charioteer
| June 8 – June 16
| Capella
| &lt;ref name=&quot;The Charioteer&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf
 |title=The Charioteer
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 17 – June 27
| Betelgeuse
| &lt;ref name=&quot;Orion&quot;/&gt;
|-
| The Dogs
| June 28 – July 7
| Sirius
| &lt;ref name=&quot;The Dogs&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf
 |title=The Dogs
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| July 8 – July 17
| Canopus
| &lt;ref name=&quot;Ship of the Argonauts&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf
 |title=The Ship of the Argonauts
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dogs
| July 18 – July 25
| Procyon
| &lt;ref name=&quot;The Dogs&quot;/&gt;
|-
| The Dragon
| July 26 – August 7
| Gianfar
| &lt;ref name=&quot;The Dragon&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 8 – August 15
| Dubhe and Merak
| &lt;ref name=&quot;The Great Bear&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&amp;date=Aug-8th@15th
 |title=The Great Bear
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Sea Serpent
| August 16 – August 23
| Alphard
| &lt;ref name=&quot;The Sea Serpent&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf
 |title=The Sea Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 24 – September 10
| Phekda, Megrez, Alioth and Mizar
| &lt;ref name=&quot;The Great Bear&quot;/&gt;
|-
| The Cup
| September 11 – September 21
| Alkes
| &lt;ref name=&quot;The Cup&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf
 |title=The Cup
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| September 22 – September 28
| Markeb
| &lt;ref name=&quot;Ship of the Argonauts&quot;/&gt;
|-
| The Raven
| September 29 – October 11
| Minkar and Algorab
| &lt;ref name=&quot;The Raven&quot;&gt;{{cite web
 |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72
 |title=The Raven
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Bear Keeper
| October 12 – October 26
| Arcturus and Izar
| &lt;ref name=&quot;The Bear Keeper&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf
 |title=The Bear Keeper
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Crown of the North Wind
| October 27 – November 10
| Alphecca
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf
 |title=The Lost Zodiac
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| November 11 – November 19
| Unuk Elhaia
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Wise Centaur
| November 20 – December 5
| Toliman (Alpha Centauri)
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf
 |title=The Wise Centaur
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Ophiuchus
| December 6 – December 16
| Han, Sabik and Ras Alhague
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf
 |title=Ophiuchus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dragon
| December 17 – December 23
| Grumium and Etanin
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| December 24 – December 28
| Alya
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|}

== Natal-chart conversion methodology ==

The Lost Zodiac proposes a proportional conversion between a conventional
zodiacal position and the corresponding Lost Zodiac period.

A placement is first treated as a position within its conventional zodiac sign.
That position is converted into a corresponding calendar date and time. The
resulting date and time are then located within the relevant Lost Zodiac period.

Each listed period includes the full span of every calendar date in its range.
The starting date begins at 12:00 a.m., and the ending date concludes at
11:59 p.m. For example, a period ending on February 28 includes the entire
day of February 28. In leap years, the Swan period ends at 11:59 p.m. on
February 29.

For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and
the end of the period as 29°59′. The beginning boundary is midnight at the
start of the first listed date, while the ending boundary is 11:59 p.m. on
the last listed date.

The proposed formula is:

&lt;math&gt;
L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59'
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; = resulting Lost Zodiac degree
* &lt;math&gt;t&lt;/math&gt; = the corresponding date and time of the natal placement
* &lt;math&gt;t_0&lt;/math&gt; = beginning of the Lost Zodiac period
* &lt;math&gt;t_1&lt;/math&gt; = end boundary of the Lost Zodiac period

=== Example ===

A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009
at approximately 6:25 AM in the example used by the author.

The Swan period runs from February 9 through the last day of February,
ending on February 28 in a common year or February 29 in a leap year.

The position of February 20 at 6:25 AM is approximately 56.34% through the
Swan period.

Therefore:

&lt;math&gt;
29^\circ 59'\times0.5634\approx16^\circ 53'
&lt;/math&gt;

The resulting proposed Lost Zodiac position is therefore:

'''Swan 16°53′'''

== Reference natal chart ==

The following conventional natal-chart placements are used as the reference
positions for the Lost Zodiac conversion.

{| class=&quot;wikitable&quot;
! Natal point
! Conventional natal position
! House
|-
| Sun
| Pisces 1°29′
| 1st House
|-
| Ascendant
| Aquarius 29°24′
| —
|-
| Moon
| Capricorn 5°24′
| 11th House
|-
| Mercury
| Aquarius 6°07′
| 12th House
|-
| Venus
| Aries 11°30′
| 2nd House
|-
| Mars
| Aquarius 11°55′
| 12th House
|-
| Jupiter
| Aquarius 10°38′
| 12th House
|-
| Saturn
| Virgo 19°41′ Rx
| 7th House
|-
| Uranus
| Pisces 21°25′
| 1st House
|-
| Neptune
| Aquarius 24°13′
| 12th House
|-
| Pluto
| Capricorn 2°47′
| 10th House
|-
| North Node
| Aquarius 8°18′ Rx
| 12th House
|-
| Lilith
| Capricorn 5°16′
| 11th House
|-
| Chiron
| Aquarius 21°52′
| 12th House
|-
| Fortune
| Aries 25°29′
| 2nd House
|-
| Vertex
| Virgo 17°15′
| 7th House
|-
| MC
| Sagittarius 8°28′
| —
|}

== Example: author's Lost Zodiac natal chart ==

Using the conversion methodology described above, the author's proposed
Lost Zodiac natal placements are:

{| class=&quot;wikitable&quot;
! Natal point
! Lost Zodiac placement
|-
| Sun
| Swan 16°53′
|-
| Ascendant
| Swan 14°06′
|-
| Moon
| Serpent 20°23′
|-
| Mercury
| Eagle 24°13′
|-
| Venus
| Pegasus 29°14′
|-
| Mars
| Dolphin 7°57′
|-
| Jupiter
| Dolphin 4°27′
|-
| Saturn
| Cup 1°52′
|-
| Uranus
| River of Night 28°32′
|-
| Neptune
| Swan 6°19′
|-
| Pluto
| Serpent 4°42′
|-
| North Node
| Eagle 28°35′
|-
| Lilith
| Serpent 19°35′
|-
| Chiron
| Swan 2°48′
|-
| Fortune
| River of Night 18°16′
|-
| Vertex
| Great Bear 27°04′
|-
| MC
| Wise Centaur 19°37′
|}</text>
      <sha1>39ndctsanku49cl6574t89kvnkbpyub</sha1>
    </revision>
    <revision>
      <id>2834699</id>
      <parentid>2834690</parentid>
      <timestamp>2026-09-27T19:36:07Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <comment>/* Background */</comment>
      <origin>2834699</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13357" sha1="biuog61buplufif3jgyzl5rkzara6ol" xml:space="preserve">{{Infobox
| title = The Lost Zodiac
| image =
| caption =

| label1 = Author
| data1 = Tahmid Arian Borno

| label2 = Year
| data2 = 2026

| label3 = Status
| data3 = Proposed hypothesis; under development

| label4 = Subject
| data4 = Astrology / stellar symbolism
}}

== The Lost Zodiac ==
'''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026.
It is an experimental astrological framework based on selected stars and
constellational figures, their associated date ranges, and a proposed method
for converting conventional natal-chart positions into Lost Zodiac positions.

&lt;div style=&quot;clear: both; width: auto; margin: 1em 0;&quot;&gt;

&lt;!-- Research Project Box --&gt;
&lt;div style=&quot;border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This is a research project at Wikiversity.'''
&lt;/div&gt;

&lt;!-- Hypothesis Warning Box --&gt;
&lt;div style=&quot;border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.'''
&lt;/div&gt;

&lt;/div&gt;

== Background ==

The Lost Zodiac is based on a set of zodiacal periods associated with prominent stars and constellational figures. The source system on which the reconstruction is based contains 31 periods representing 22 distinct constellational figures. Several figures recur in non-adjacent periods because different stars associated with the same figure are assigned to different parts of the year.

The Lost Zodiac system used as the basis for this project was created by
astrologer Catherine Tennant and features 22 &quot;lost&quot; star signs based on
major constellations located north and south of the main zodiac
belt.&lt;ref name=&quot;Interactive Stars&quot;&gt;{{cite web
|url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php
|title=The Lost Zodiac
|publisher=InteractiveStars.com
|access-date=2026-09-23
}}&lt;/ref&gt; The period boundaries, associated figures, and principal stars
used in this project are derived directly from that source; the
organization of the periods within this article and the natal-chart
conversion methodology described below remain original to the author.

The source material describes the system as a set of &quot;lost&quot; star signs based on major constellations located north and south of the main zodiac belt.&lt;ref name=&quot;InteractiveStars&quot;/&gt;

The present project does not claim that the organization of these periods or their relationship to the modern natal zodiac has been historically established. Instead, the source material is used as the astronomical and symbolic foundation for a proposed reconstruction.

== The Lost Zodiac Signs ==

{| class=&quot;wikitable sortable&quot;
! Sign
! Period
! Associated star(s)
! Sources
|-
| The Lyre of Orpheus
| December 29 – January 13
| Vega
| &lt;ref name=&quot;The Lyre of Orpheus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&amp;date=Dec-29th@Jan-13th
 |title=The Lyre of Orpheus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Eagle
| January 14 – January 28
| Altair
| &lt;ref name=&quot;The Eagle&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf
 |title=The Eagle
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dolphin
| January 29 – February 8
| Rotanev and Sualocin
| &lt;ref name=&quot;The Dolphin&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&amp;date=Jan-29th@Feb-8th
 |title=The Dolphin
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Swan
| February 9 – February 28/29
| Sadir
| &lt;ref name=&quot;The Swan&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf
 |title=The Swan
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| March 1 – March 12
| Achernar
| &lt;ref name=&quot;The River of Night&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf
 |title=The River of Night
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Pegasus
| March 13 – April 1
| Markab, Scheat and Algenib
| &lt;ref name=&quot;Pegasus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf
 |title=Pegasus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Andromeda
| April 2 – April 9
| Alpheratz
| &lt;ref name=&quot;Andromeda&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf
 |title=Andromeda
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| April 10 – April 18
| Acamar
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Andromeda
| April 19 – May 8
| Mirach and Alamach
| &lt;ref name=&quot;Andromeda&quot;/&gt;
|-
| The River of Night
| May 9 – May 15
| Rana and Zanrak
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Perseus
| May 16 – May 31
| Algol and Mirfak
| &lt;ref name=&quot;Perseus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&amp;date=May16th@31st
 |title=Perseus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 1 – June 7
| Rigel
| &lt;ref name=&quot;Orion&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf
 |title=Orion
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Charioteer
| June 8 – June 16
| Capella
| &lt;ref name=&quot;The Charioteer&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf
 |title=The Charioteer
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 17 – June 27
| Betelgeuse
| &lt;ref name=&quot;Orion&quot;/&gt;
|-
| The Dogs
| June 28 – July 7
| Sirius
| &lt;ref name=&quot;The Dogs&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf
 |title=The Dogs
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| July 8 – July 17
| Canopus
| &lt;ref name=&quot;Ship of the Argonauts&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf
 |title=The Ship of the Argonauts
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dogs
| July 18 – July 25
| Procyon
| &lt;ref name=&quot;The Dogs&quot;/&gt;
|-
| The Dragon
| July 26 – August 7
| Gianfar
| &lt;ref name=&quot;The Dragon&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 8 – August 15
| Dubhe and Merak
| &lt;ref name=&quot;The Great Bear&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&amp;date=Aug-8th@15th
 |title=The Great Bear
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Sea Serpent
| August 16 – August 23
| Alphard
| &lt;ref name=&quot;The Sea Serpent&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf
 |title=The Sea Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 24 – September 10
| Phekda, Megrez, Alioth and Mizar
| &lt;ref name=&quot;The Great Bear&quot;/&gt;
|-
| The Cup
| September 11 – September 21
| Alkes
| &lt;ref name=&quot;The Cup&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf
 |title=The Cup
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| September 22 – September 28
| Markeb
| &lt;ref name=&quot;Ship of the Argonauts&quot;/&gt;
|-
| The Raven
| September 29 – October 11
| Minkar and Algorab
| &lt;ref name=&quot;The Raven&quot;&gt;{{cite web
 |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72
 |title=The Raven
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Bear Keeper
| October 12 – October 26
| Arcturus and Izar
| &lt;ref name=&quot;The Bear Keeper&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf
 |title=The Bear Keeper
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Crown of the North Wind
| October 27 – November 10
| Alphecca
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf
 |title=The Lost Zodiac
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| November 11 – November 19
| Unuk Elhaia
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Wise Centaur
| November 20 – December 5
| Toliman (Alpha Centauri)
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf
 |title=The Wise Centaur
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Ophiuchus
| December 6 – December 16
| Han, Sabik and Ras Alhague
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf
 |title=Ophiuchus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dragon
| December 17 – December 23
| Grumium and Etanin
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| December 24 – December 28
| Alya
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|}

== Natal-chart conversion methodology ==

The Lost Zodiac proposes a proportional conversion between a conventional
zodiacal position and the corresponding Lost Zodiac period.

A placement is first treated as a position within its conventional zodiac sign.
That position is converted into a corresponding calendar date and time. The
resulting date and time are then located within the relevant Lost Zodiac period.

Each listed period includes the full span of every calendar date in its range.
The starting date begins at 12:00 a.m., and the ending date concludes at
11:59 p.m. For example, a period ending on February 28 includes the entire
day of February 28. In leap years, the Swan period ends at 11:59 p.m. on
February 29.

For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and
the end of the period as 29°59′. The beginning boundary is midnight at the
start of the first listed date, while the ending boundary is 11:59 p.m. on
the last listed date.

The proposed formula is:

&lt;math&gt;
L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59'
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; = resulting Lost Zodiac degree
* &lt;math&gt;t&lt;/math&gt; = the corresponding date and time of the natal placement
* &lt;math&gt;t_0&lt;/math&gt; = beginning of the Lost Zodiac period
* &lt;math&gt;t_1&lt;/math&gt; = end boundary of the Lost Zodiac period

=== Example ===

A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009
at approximately 6:25 AM in the example used by the author.

The Swan period runs from February 9 through the last day of February,
ending on February 28 in a common year or February 29 in a leap year.

The position of February 20 at 6:25 AM is approximately 56.34% through the
Swan period.

Therefore:

&lt;math&gt;
29^\circ 59'\times0.5634\approx16^\circ 53'
&lt;/math&gt;

The resulting proposed Lost Zodiac position is therefore:

'''Swan 16°53′'''

== Reference natal chart ==

The following conventional natal-chart placements are used as the reference
positions for the Lost Zodiac conversion.

{| class=&quot;wikitable&quot;
! Natal point
! Conventional natal position
! House
|-
| Sun
| Pisces 1°29′
| 1st House
|-
| Ascendant
| Aquarius 29°24′
| —
|-
| Moon
| Capricorn 5°24′
| 11th House
|-
| Mercury
| Aquarius 6°07′
| 12th House
|-
| Venus
| Aries 11°30′
| 2nd House
|-
| Mars
| Aquarius 11°55′
| 12th House
|-
| Jupiter
| Aquarius 10°38′
| 12th House
|-
| Saturn
| Virgo 19°41′ Rx
| 7th House
|-
| Uranus
| Pisces 21°25′
| 1st House
|-
| Neptune
| Aquarius 24°13′
| 12th House
|-
| Pluto
| Capricorn 2°47′
| 10th House
|-
| North Node
| Aquarius 8°18′ Rx
| 12th House
|-
| Lilith
| Capricorn 5°16′
| 11th House
|-
| Chiron
| Aquarius 21°52′
| 12th House
|-
| Fortune
| Aries 25°29′
| 2nd House
|-
| Vertex
| Virgo 17°15′
| 7th House
|-
| MC
| Sagittarius 8°28′
| —
|}

== Example: author's Lost Zodiac natal chart ==

Using the conversion methodology described above, the author's proposed
Lost Zodiac natal placements are:

{| class=&quot;wikitable&quot;
! Natal point
! Lost Zodiac placement
|-
| Sun
| Swan 16°53′
|-
| Ascendant
| Swan 14°06′
|-
| Moon
| Serpent 20°23′
|-
| Mercury
| Eagle 24°13′
|-
| Venus
| Pegasus 29°14′
|-
| Mars
| Dolphin 7°57′
|-
| Jupiter
| Dolphin 4°27′
|-
| Saturn
| Cup 1°52′
|-
| Uranus
| River of Night 28°32′
|-
| Neptune
| Swan 6°19′
|-
| Pluto
| Serpent 4°42′
|-
| North Node
| Eagle 28°35′
|-
| Lilith
| Serpent 19°35′
|-
| Chiron
| Swan 2°48′
|-
| Fortune
| River of Night 18°16′
|-
| Vertex
| Great Bear 27°04′
|-
| MC
| Wise Centaur 19°37′
|}</text>
      <sha1>biuog61buplufif3jgyzl5rkzara6ol</sha1>
    </revision>
    <revision>
      <id>2834700</id>
      <parentid>2834699</parentid>
      <timestamp>2026-09-27T19:39:12Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <origin>2834700</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13362" sha1="c8zkwx7dftraf1wl1ng0dlatt8ohwnf" xml:space="preserve">{{Infobox
| title = The Lost Zodiac
| image =
| caption =

| label1 = Author
| data1 = Tahmid Arian Borno

| label2 = Year
| data2 = 2026

| label3 = Status
| data3 = Proposed hypothesis; under development

| label4 = Subject
| data4 = Astrology / stellar symbolism
}}

== The Lost Zodiac ==
'''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026.
It is an experimental astrological framework based on selected stars and
constellational figures, their associated date ranges, and a proposed method
for converting conventional natal-chart positions into Lost Zodiac positions.

&lt;div style=&quot;clear: both; width: auto; margin: 1em 0;&quot;&gt;

&lt;!-- Research Project Box --&gt;
&lt;div style=&quot;border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This is a research project at Wikiversity.'''
&lt;/div&gt;

&lt;!-- Hypothesis Warning Box --&gt;
&lt;div style=&quot;border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.'''
&lt;/div&gt;

&lt;/div&gt;

== Background ==

The Lost Zodiac is based on a set of zodiacal periods associated with prominent stars and constellational figures. The source system on which the reconstruction is based contains 31 periods representing 22 distinct constellational figures. Several figures recur in non-adjacent periods because different stars associated with the same figure are assigned to different parts of the year.

The Lost Zodiac system used as the basis for this project was created by
astrologer Catherine Tennant and features 22 &quot;lost&quot; star signs based on
major constellations located north and south of the main zodiac
belt.&lt;ref name=&quot;Interactive Stars&quot;&gt;{{cite web
|url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac_sign.php
|title=The Lost Zodiac
|publisher=InteractiveStars.com
|access-date=2026-09-23
}}&lt;/ref&gt; The period boundaries, associated figures, and principal stars
used in this project are derived directly from that source; the
organization of the periods within this article and the natal-chart
conversion methodology described below remain original to the author.

The source material describes the system as a set of &quot;lost&quot; star signs based on major constellations located north and south of the main zodiac belt.&lt;ref name=&quot;InteractiveStars&quot;/&gt;

The present project does not claim that the organization of these periods or their relationship to the modern natal zodiac has been historically established. Instead, the source material is used as the astronomical and symbolic foundation for a proposed reconstruction.

== The Lost Zodiac Signs ==

{| class=&quot;wikitable sortable&quot;
! Sign
! Period
! Associated star(s)
! Sources
|-
| The Lyre of Orpheus
| December 29 – January 13
| Vega
| &lt;ref name=&quot;The Lyre of Orpheus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&amp;date=Dec-29th@Jan-13th
 |title=The Lyre of Orpheus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Eagle
| January 14 – January 28
| Altair
| &lt;ref name=&quot;The Eagle&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf
 |title=The Eagle
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dolphin
| January 29 – February 8
| Rotanev and Sualocin
| &lt;ref name=&quot;The Dolphin&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&amp;date=Jan-29th@Feb-8th
 |title=The Dolphin
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Swan
| February 9 – February 28/29
| Sadir
| &lt;ref name=&quot;The Swan&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf
 |title=The Swan
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| March 1 – March 12
| Achernar
| &lt;ref name=&quot;The River of Night&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf
 |title=The River of Night
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Pegasus
| March 13 – April 1
| Markab, Scheat and Algenib
| &lt;ref name=&quot;Pegasus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf
 |title=Pegasus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Andromeda
| April 2 – April 9
| Alpheratz
| &lt;ref name=&quot;Andromeda&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf
 |title=Andromeda
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| April 10 – April 18
| Acamar
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Andromeda
| April 19 – May 8
| Mirach and Alamach
| &lt;ref name=&quot;Andromeda&quot;/&gt;
|-
| The River of Night
| May 9 – May 15
| Rana and Zanrak
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Perseus
| May 16 – May 31
| Algol and Mirfak
| &lt;ref name=&quot;Perseus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&amp;date=May16th@31st
 |title=Perseus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 1 – June 7
| Rigel
| &lt;ref name=&quot;Orion&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf
 |title=Orion
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Charioteer
| June 8 – June 16
| Capella
| &lt;ref name=&quot;The Charioteer&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf
 |title=The Charioteer
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 17 – June 27
| Betelgeuse
| &lt;ref name=&quot;Orion&quot;/&gt;
|-
| The Dogs
| June 28 – July 7
| Sirius
| &lt;ref name=&quot;The Dogs&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf
 |title=The Dogs
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| July 8 – July 17
| Canopus
| &lt;ref name=&quot;Ship of the Argonauts&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf
 |title=The Ship of the Argonauts
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dogs
| July 18 – July 25
| Procyon
| &lt;ref name=&quot;The Dogs&quot;/&gt;
|-
| The Dragon
| July 26 – August 7
| Gianfar
| &lt;ref name=&quot;The Dragon&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 8 – August 15
| Dubhe and Merak
| &lt;ref name=&quot;The Great Bear&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&amp;date=Aug-8th@15th
 |title=The Great Bear
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Sea Serpent
| August 16 – August 23
| Alphard
| &lt;ref name=&quot;The Sea Serpent&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf
 |title=The Sea Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 24 – September 10
| Phekda, Megrez, Alioth and Mizar
| &lt;ref name=&quot;The Great Bear&quot;/&gt;
|-
| The Cup
| September 11 – September 21
| Alkes
| &lt;ref name=&quot;The Cup&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf
 |title=The Cup
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| September 22 – September 28
| Markeb
| &lt;ref name=&quot;Ship of the Argonauts&quot;/&gt;
|-
| The Raven
| September 29 – October 11
| Minkar and Algorab
| &lt;ref name=&quot;The Raven&quot;&gt;{{cite web
 |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72
 |title=The Raven
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Bear Keeper
| October 12 – October 26
| Arcturus and Izar
| &lt;ref name=&quot;The Bear Keeper&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf
 |title=The Bear Keeper
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Crown of the North Wind
| October 27 – November 10
| Alphecca
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf
 |title=The Lost Zodiac
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| November 11 – November 19
| Unuk Elhaia
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Wise Centaur
| November 20 – December 5
| Toliman (Alpha Centauri)
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf
 |title=The Wise Centaur
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Ophiuchus
| December 6 – December 16
| Han, Sabik and Ras Alhague
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf
 |title=Ophiuchus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dragon
| December 17 – December 23
| Grumium and Etanin
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| December 24 – December 28
| Alya
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|}

== Natal-chart conversion methodology ==

The Lost Zodiac proposes a proportional conversion between a conventional
zodiacal position and the corresponding Lost Zodiac period.

A placement is first treated as a position within its conventional zodiac sign.
That position is converted into a corresponding calendar date and time. The
resulting date and time are then located within the relevant Lost Zodiac period.

Each listed period includes the full span of every calendar date in its range.
The starting date begins at 12:00 a.m., and the ending date concludes at
11:59 p.m. For example, a period ending on February 28 includes the entire
day of February 28. In leap years, the Swan period ends at 11:59 p.m. on
February 29.

For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and
the end of the period as 29°59′. The beginning boundary is midnight at the
start of the first listed date, while the ending boundary is 11:59 p.m. on
the last listed date.

The proposed formula is:

&lt;math&gt;
L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59'
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; = resulting Lost Zodiac degree
* &lt;math&gt;t&lt;/math&gt; = the corresponding date and time of the natal placement
* &lt;math&gt;t_0&lt;/math&gt; = beginning of the Lost Zodiac period
* &lt;math&gt;t_1&lt;/math&gt; = end boundary of the Lost Zodiac period

=== Example ===

A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009
at approximately 6:25 AM in the example used by the author.

The Swan period runs from February 9 through the last day of February,
ending on February 28 in a common year or February 29 in a leap year.

The position of February 20 at 6:25 AM is approximately 56.34% through the
Swan period.

Therefore:

&lt;math&gt;
29^\circ 59'\times0.5634\approx16^\circ 53'
&lt;/math&gt;

The resulting proposed Lost Zodiac position is therefore:

'''Swan 16°53′'''

== Reference natal chart ==

The following conventional natal-chart placements are used as the reference
positions for the Lost Zodiac conversion.

{| class=&quot;wikitable&quot;
! Natal point
! Conventional natal position
! House
|-
| Sun
| Pisces 1°29′
| 1st House
|-
| Ascendant
| Aquarius 29°24′
| —
|-
| Moon
| Capricorn 5°24′
| 11th House
|-
| Mercury
| Aquarius 6°07′
| 12th House
|-
| Venus
| Aries 11°30′
| 2nd House
|-
| Mars
| Aquarius 11°55′
| 12th House
|-
| Jupiter
| Aquarius 10°38′
| 12th House
|-
| Saturn
| Virgo 19°41′ Rx
| 7th House
|-
| Uranus
| Pisces 21°25′
| 1st House
|-
| Neptune
| Aquarius 24°13′
| 12th House
|-
| Pluto
| Capricorn 2°47′
| 10th House
|-
| North Node
| Aquarius 8°18′ Rx
| 12th House
|-
| Lilith
| Capricorn 5°16′
| 11th House
|-
| Chiron
| Aquarius 21°52′
| 12th House
|-
| Fortune
| Aries 25°29′
| 2nd House
|-
| Vertex
| Virgo 17°15′
| 7th House
|-
| MC
| Sagittarius 8°28′
| —
|}

== Example: author's Lost Zodiac natal chart ==

Using the conversion methodology described above, the author's proposed
Lost Zodiac natal placements are:

{| class=&quot;wikitable&quot;
! Natal point
! Lost Zodiac placement
|-
| Sun
| Swan 16°53′
|-
| Ascendant
| Swan 14°06′
|-
| Moon
| Serpent 20°23′
|-
| Mercury
| Eagle 24°13′
|-
| Venus
| Pegasus 29°14′
|-
| Mars
| Dolphin 7°57′
|-
| Jupiter
| Dolphin 4°27′
|-
| Saturn
| Cup 1°52′
|-
| Uranus
| River of Night 28°32′
|-
| Neptune
| Swan 6°19′
|-
| Pluto
| Serpent 4°42′
|-
| North Node
| Eagle 28°35′
|-
| Lilith
| Serpent 19°35′
|-
| Chiron
| Swan 2°48′
|-
| Fortune
| River of Night 18°16′
|-
| Vertex
| Great Bear 27°04′
|-
| MC
| Wise Centaur 19°37′
|}</text>
      <sha1>c8zkwx7dftraf1wl1ng0dlatt8ohwnf</sha1>
    </revision>
    <revision>
      <id>2834701</id>
      <parentid>2834700</parentid>
      <timestamp>2026-09-27T19:50:48Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <comment>/* Background */</comment>
      <origin>2834701</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13181" sha1="5tzh2f8aj49r4y0dbe9u203zw3kfznf" xml:space="preserve">{{Infobox
| title = The Lost Zodiac
| image =
| caption =

| label1 = Author
| data1 = Tahmid Arian Borno

| label2 = Year
| data2 = 2026

| label3 = Status
| data3 = Proposed hypothesis; under development

| label4 = Subject
| data4 = Astrology / stellar symbolism
}}

== The Lost Zodiac ==
'''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026.
It is an experimental astrological framework based on selected stars and
constellational figures, their associated date ranges, and a proposed method
for converting conventional natal-chart positions into Lost Zodiac positions.

&lt;div style=&quot;clear: both; width: auto; margin: 1em 0;&quot;&gt;

&lt;!-- Research Project Box --&gt;
&lt;div style=&quot;border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This is a research project at Wikiversity.'''
&lt;/div&gt;

&lt;!-- Hypothesis Warning Box --&gt;
&lt;div style=&quot;border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;&quot;&gt;
'''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.'''
&lt;/div&gt;

&lt;/div&gt;

== Background ==

The Lost Zodiac is based on a set of zodiacal periods associated with prominent stars and constellational figures. The source system on which the reconstruction is based contains 31 periods representing 22 distinct constellational figures. Several figures recur in non-adjacent periods because different stars associated with the same figure are assigned to different parts of the year.

The Lost Zodiac system used as the basis for this project was created by
astrologer Catherine Tennant and features 22 &quot;lost&quot; star signs based on
major constellations located north and south of the main zodiac
belt.&lt;ref name=&quot;Interactive Stars&quot;&gt;{{cite web
|url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac_sign.php
|title=The Lost Zodiac
|publisher=InteractiveStars.com
|access-date=2026-09-23
}}&lt;/ref&gt; The period boundaries, associated figures, and principal stars
used in this project are derived directly from that source; the
organization of the periods within this article and the natal-chart
conversion methodology described below remain original to the author.

The present project does not claim that the organization of these periods or their relationship to the modern natal zodiac has been historically established. Instead, the source material is used as the astronomical and symbolic foundation for a proposed reconstruction.

== The Lost Zodiac Signs ==

{| class=&quot;wikitable sortable&quot;
! Sign
! Period
! Associated star(s)
! Sources
|-
| The Lyre of Orpheus
| December 29 – January 13
| Vega
| &lt;ref name=&quot;The Lyre of Orpheus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&amp;date=Dec-29th@Jan-13th
 |title=The Lyre of Orpheus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Eagle
| January 14 – January 28
| Altair
| &lt;ref name=&quot;The Eagle&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf
 |title=The Eagle
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dolphin
| January 29 – February 8
| Rotanev and Sualocin
| &lt;ref name=&quot;The Dolphin&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&amp;date=Jan-29th@Feb-8th
 |title=The Dolphin
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Swan
| February 9 – February 28/29
| Sadir
| &lt;ref name=&quot;The Swan&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf
 |title=The Swan
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| March 1 – March 12
| Achernar
| &lt;ref name=&quot;The River of Night&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf
 |title=The River of Night
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Pegasus
| March 13 – April 1
| Markab, Scheat and Algenib
| &lt;ref name=&quot;Pegasus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf
 |title=Pegasus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Andromeda
| April 2 – April 9
| Alpheratz
| &lt;ref name=&quot;Andromeda&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf
 |title=Andromeda
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The River of Night
| April 10 – April 18
| Acamar
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Andromeda
| April 19 – May 8
| Mirach and Alamach
| &lt;ref name=&quot;Andromeda&quot;/&gt;
|-
| The River of Night
| May 9 – May 15
| Rana and Zanrak
| &lt;ref name=&quot;The River of Night&quot;/&gt;
|-
| Perseus
| May 16 – May 31
| Algol and Mirfak
| &lt;ref name=&quot;Perseus&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&amp;date=May16th@31st
 |title=Perseus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 1 – June 7
| Rigel
| &lt;ref name=&quot;Orion&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf
 |title=Orion
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Charioteer
| June 8 – June 16
| Capella
| &lt;ref name=&quot;The Charioteer&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf
 |title=The Charioteer
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Orion
| June 17 – June 27
| Betelgeuse
| &lt;ref name=&quot;Orion&quot;/&gt;
|-
| The Dogs
| June 28 – July 7
| Sirius
| &lt;ref name=&quot;The Dogs&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf
 |title=The Dogs
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| July 8 – July 17
| Canopus
| &lt;ref name=&quot;Ship of the Argonauts&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf
 |title=The Ship of the Argonauts
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dogs
| July 18 – July 25
| Procyon
| &lt;ref name=&quot;The Dogs&quot;/&gt;
|-
| The Dragon
| July 26 – August 7
| Gianfar
| &lt;ref name=&quot;The Dragon&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 8 – August 15
| Dubhe and Merak
| &lt;ref name=&quot;The Great Bear&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&amp;date=Aug-8th@15th
 |title=The Great Bear
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Sea Serpent
| August 16 – August 23
| Alphard
| &lt;ref name=&quot;The Sea Serpent&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf
 |title=The Sea Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Great Bear
| August 24 – September 10
| Phekda, Megrez, Alioth and Mizar
| &lt;ref name=&quot;The Great Bear&quot;/&gt;
|-
| The Cup
| September 11 – September 21
| Alkes
| &lt;ref name=&quot;The Cup&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf
 |title=The Cup
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Ship of the Argonauts
| September 22 – September 28
| Markeb
| &lt;ref name=&quot;Ship of the Argonauts&quot;/&gt;
|-
| The Raven
| September 29 – October 11
| Minkar and Algorab
| &lt;ref name=&quot;The Raven&quot;&gt;{{cite web
 |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72
 |title=The Raven
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Bear Keeper
| October 12 – October 26
| Arcturus and Izar
| &lt;ref name=&quot;The Bear Keeper&quot;&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf
 |title=The Bear Keeper
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Crown of the North Wind
| October 27 – November 10
| Alphecca
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf
 |title=The Lost Zodiac
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| November 11 – November 19
| Unuk Elhaia
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Wise Centaur
| November 20 – December 5
| Toliman (Alpha Centauri)
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf
 |title=The Wise Centaur
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| Ophiuchus
| December 6 – December 16
| Han, Sabik and Ras Alhague
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf
 |title=Ophiuchus
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Dragon
| December 17 – December 23
| Grumium and Etanin
| &lt;ref&gt;{{cite web
 |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf
 |title=The Dragon
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|-
| The Serpent
| December 24 – December 28
| Alya
| &lt;ref&gt;{{cite web
 |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ
 |title=The Serpent
 |publisher=InteractiveStars.com
 |access-date=2026-09-24
}}&lt;/ref&gt;
|}

== Natal-chart conversion methodology ==

The Lost Zodiac proposes a proportional conversion between a conventional
zodiacal position and the corresponding Lost Zodiac period.

A placement is first treated as a position within its conventional zodiac sign.
That position is converted into a corresponding calendar date and time. The
resulting date and time are then located within the relevant Lost Zodiac period.

Each listed period includes the full span of every calendar date in its range.
The starting date begins at 12:00 a.m., and the ending date concludes at
11:59 p.m. For example, a period ending on February 28 includes the entire
day of February 28. In leap years, the Swan period ends at 11:59 p.m. on
February 29.

For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and
the end of the period as 29°59′. The beginning boundary is midnight at the
start of the first listed date, while the ending boundary is 11:59 p.m. on
the last listed date.

The proposed formula is:

&lt;math&gt;
L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59'
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; = resulting Lost Zodiac degree
* &lt;math&gt;t&lt;/math&gt; = the corresponding date and time of the natal placement
* &lt;math&gt;t_0&lt;/math&gt; = beginning of the Lost Zodiac period
* &lt;math&gt;t_1&lt;/math&gt; = end boundary of the Lost Zodiac period

=== Example ===

A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009
at approximately 6:25 AM in the example used by the author.

The Swan period runs from February 9 through the last day of February,
ending on February 28 in a common year or February 29 in a leap year.

The position of February 20 at 6:25 AM is approximately 56.34% through the
Swan period.

Therefore:

&lt;math&gt;
29^\circ 59'\times0.5634\approx16^\circ 53'
&lt;/math&gt;

The resulting proposed Lost Zodiac position is therefore:

'''Swan 16°53′'''

== Reference natal chart ==

The following conventional natal-chart placements are used as the reference
positions for the Lost Zodiac conversion.

{| class=&quot;wikitable&quot;
! Natal point
! Conventional natal position
! House
|-
| Sun
| Pisces 1°29′
| 1st House
|-
| Ascendant
| Aquarius 29°24′
| —
|-
| Moon
| Capricorn 5°24′
| 11th House
|-
| Mercury
| Aquarius 6°07′
| 12th House
|-
| Venus
| Aries 11°30′
| 2nd House
|-
| Mars
| Aquarius 11°55′
| 12th House
|-
| Jupiter
| Aquarius 10°38′
| 12th House
|-
| Saturn
| Virgo 19°41′ Rx
| 7th House
|-
| Uranus
| Pisces 21°25′
| 1st House
|-
| Neptune
| Aquarius 24°13′
| 12th House
|-
| Pluto
| Capricorn 2°47′
| 10th House
|-
| North Node
| Aquarius 8°18′ Rx
| 12th House
|-
| Lilith
| Capricorn 5°16′
| 11th House
|-
| Chiron
| Aquarius 21°52′
| 12th House
|-
| Fortune
| Aries 25°29′
| 2nd House
|-
| Vertex
| Virgo 17°15′
| 7th House
|-
| MC
| Sagittarius 8°28′
| —
|}

== Example: author's Lost Zodiac natal chart ==

Using the conversion methodology described above, the author's proposed
Lost Zodiac natal placements are:

{| class=&quot;wikitable&quot;
! Natal point
! Lost Zodiac placement
|-
| Sun
| Swan 16°53′
|-
| Ascendant
| Swan 14°06′
|-
| Moon
| Serpent 20°23′
|-
| Mercury
| Eagle 24°13′
|-
| Venus
| Pegasus 29°14′
|-
| Mars
| Dolphin 7°57′
|-
| Jupiter
| Dolphin 4°27′
|-
| Saturn
| Cup 1°52′
|-
| Uranus
| River of Night 28°32′
|-
| Neptune
| Swan 6°19′
|-
| Pluto
| Serpent 4°42′
|-
| North Node
| Eagle 28°35′
|-
| Lilith
| Serpent 19°35′
|-
| Chiron
| Swan 2°48′
|-
| Fortune
| River of Night 18°16′
|-
| Vertex
| Great Bear 27°04′
|-
| MC
| Wise Centaur 19°37′
|}</text>
      <sha1>5tzh2f8aj49r4y0dbe9u203zw3kfznf</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Huisartsen1</title>
    <ns>3</ns>
    <id>332244</id>
    <revision>
      <id>2834679</id>
      <parentid>2834633</parentid>
      <timestamp>2026-09-27T15:01:50Z</timestamp>
      <contributor>
        <username>Huisartsen1</username>
        <id>3111673</id>
      </contributor>
      <comment>/* Understanding the Importance of Regular Health Check-Ups */ new section</comment>
      <origin>2834679</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5004" sha1="601z4sosjdbl4wueh77z2fhojr5w78b" xml:space="preserve">

== 2026-09-27 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Huisartsen1|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1790482830337.6s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 04:20, 27 September 2026 (UTC)

== Understanding the Importance of Regular Health Check-Ups ==
Regular health check-ups are an important part of maintaining overall health and well-being. Many health conditions can develop gradually without causing noticeable symptoms in their early stages. Routine medical consultations give people an opportunity to discuss concerns, '''''&lt;u&gt;[https://huisartsenkruisdorp.nl/ monitor existing health issues]&lt;/u&gt;''''', and receive appropriate advice about prevention and healthy lifestyle choices.

== Why Regular Check-Ups Matter ==
A health check-up is not only useful when someone feels sick. A doctor can discuss changes in symptoms, lifestyle, family medical history, medications, and other factors that may affect long-term health. Depending on a person's age, medical history, and individual circumstances, a healthcare professional may recommend specific examinations or tests.

Early attention can be particularly valuable for conditions such as high blood pressure, diabetes, cardiovascular disease, and certain types of cancer. Some of these conditions may remain unnoticed for a considerable period of time. Identifying potential problems earlier can allow patients and healthcare professionals to discuss appropriate next steps.

== Monitoring Blood Pressure and Other Risk Factors ==
Blood pressure is one example of a health measurement that can provide useful information even when a person feels completely well. Persistently high blood pressure can increase the risk of cardiovascular and other health complications. Regular monitoring can help identify unusual readings and provide an opportunity to discuss lifestyle changes or further medical evaluation.

Other factors may also be considered during a consultation, including weight, smoking, alcohol consumption, physical activity, diet, and family history. Looking at these factors together can provide a broader picture of a person's health.

== Prevention Through Healthy Habits ==
Medical care is only one part of maintaining good health. Everyday habits can also have a significant role. Eating a balanced diet, staying physically active, getting sufficient sleep, avoiding tobacco, and maintaining a healthy weight are commonly recommended ways to support general health.

Small and sustainable changes are often easier to maintain than major changes made for a short period. People who are unsure which lifestyle changes are appropriate for them can discuss their circumstances with a qualified healthcare professional.

== When to Contact a Doctor ==
People should not necessarily wait for their next routine appointment if they develop a new or concerning symptom. Persistent pain, unexplained weight changes, unusual bleeding, prolonged fatigue, breathing difficulties, or other significant changes in health may require medical assessment.

The appropriate response depends on the symptoms and their severity. In an emergency, immediate emergency medical services should be contacted rather than waiting for a routine appointment.

== Keeping Track of Your Health ==
Keeping a record of important health information can make medical consultations more useful. Patients may wish to keep track of current medications, allergies, previous diagnoses, vaccinations, family medical history, and recurring symptoms.

Writing down questions before an appointment can also help ensure that important concerns are discussed. Clear communication between patients and healthcare professionals supports better understanding and informed healthcare decisions.

== Conclusion ==
Regular health check-ups can help people stay informed about their health and identify potential concerns that may otherwise go unnoticed. Combined with healthy lifestyle habits and appropriate medical attention when symptoms occur, routine healthcare can contribute to long-term well-being.

Every person's healthcare needs are different. For personalised advice, screening recommendations, or concerns about a particular symptom, it is best to consult a qualified healthcare professional. [[User:Huisartsen1|Huisartsen1]] ([[User talk:Huisartsen1|discuss]] • [[Special:Contributions/Huisartsen1|contribs]]) 15:01, 27 September 2026 (UTC)</text>
      <sha1>601z4sosjdbl4wueh77z2fhojr5w78b</sha1>
    </revision>
    <revision>
      <id>2834680</id>
      <parentid>2834679</parentid>
      <timestamp>2026-09-27T15:03:06Z</timestamp>
      <contributor>
        <username>Veritas Sapientiae</username>
        <id>2951636</id>
      </contributor>
      <minor />
      <comment>Reverted edits by [[Special:Contribs/Huisartsen1|Huisartsen1]] ([[User talk:Huisartsen1|talk]]) to last version by MathXplore: unnecessary links or spam</comment>
      <origin>2834633</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="812" sha1="062309qawa87uu5np08fsmyz1438ajs" xml:space="preserve">

== 2026-09-27 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Huisartsen1|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1790482830337.6s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 04:20, 27 September 2026 (UTC)</text>
      <sha1>062309qawa87uu5np08fsmyz1438ajs</sha1>
    </revision>
  </page>
  <page>
    <title>User:202 Xup iartkab lakarg</title>
    <ns>2</ns>
    <id>332246</id>
    <redirect title="The Lost Zodiac" />
    <revision>
      <id>2834683</id>
      <timestamp>2026-09-27T15:57:02Z</timestamp>
      <contributor>
        <username>202 Xup iartkab lakarg</username>
        <id>3111500</id>
      </contributor>
      <comment>202 Xup iartkab lakarg moved page [[User:202 Xup iartkab lakarg]] to [[The Lost Zodiac]]</comment>
      <origin>2834683</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="29" sha1="9hltsgpyx3wcs2as20wuy7icvotpiog" xml:space="preserve">#REDIRECT [[The Lost Zodiac]]</text>
      <sha1>9hltsgpyx3wcs2as20wuy7icvotpiog</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Petrosa51</title>
    <ns>3</ns>
    <id>332247</id>
    <revision>
      <id>2834684</id>
      <timestamp>2026-09-27T16:24:08Z</timestamp>
      <contributor>
        <username>Koavf</username>
        <id>147</id>
      </contributor>
      <comment>Created page with &quot;{{subst:welcome}} ~~~~&quot;</comment>
      <origin>2834684</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2709" sha1="iv5b4evkjue6cmgbw3j04crzyctksu4" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Petrosa51!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Koavf|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.

To find your way around, check out:
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;

* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
* Learn [[Help:How to write an educational resource|how to write an educational resource]]
* Find out about [[Wikiversity:Research|research]] activities
* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! --―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:24, 27 September 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:24, 27 September 2026 (UTC)</text>
      <sha1>iv5b4evkjue6cmgbw3j04crzyctksu4</sha1>
    </revision>
  </page>
  <page>
    <title>Peter Hopley</title>
    <ns>0</ns>
    <id>332248</id>
    <revision>
      <id>2834691</id>
      <timestamp>2026-09-27T18:43:55Z</timestamp>
      <contributor>
        <username>~2026-52092-05</username>
        <id>3111720</id>
      </contributor>
      <comment>Created page with &quot;Peter John Kames Hopley, 11th oct 2026 will be 53 years&quot;</comment>
      <origin>2834691</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55" sha1="38lvtg5k5vee2ws2z4b3knqb185yzm3" xml:space="preserve">Peter John Kames Hopley, 11th oct 2026 will be 53 years</text>
      <sha1>38lvtg5k5vee2ws2z4b3knqb185yzm3</sha1>
    </revision>
    <revision>
      <id>2834692</id>
      <parentid>2834691</parentid>
      <timestamp>2026-09-27T18:45:22Z</timestamp>
      <contributor>
        <username>~2026-52092-05</username>
        <id>3111720</id>
      </contributor>
      <origin>2834692</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="76" sha1="l4on1l7exkf0jjhvywd5zuqxp3evpa8" xml:space="preserve">Peter John Kames Hopley, 11th oct 2026 will be 53 years.  Software Developer</text>
      <sha1>l4on1l7exkf0jjhvywd5zuqxp3evpa8</sha1>
    </revision>
    <revision>
      <id>2834693</id>
      <parentid>2834692</parentid>
      <timestamp>2026-09-27T18:46:04Z</timestamp>
      <contributor>
        <username>~2026-52092-05</username>
        <id>3111720</id>
      </contributor>
      <comment>added my website</comment>
      <origin>2834693</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="104" sha1="fb1i83wky9do8wq72of3ctk8dm91qh0" xml:space="preserve">Peter John Kames Hopley, 11th oct 2026 will be 53 years.  Software Developer.  Owns www,vistamations.com</text>
      <sha1>fb1i83wky9do8wq72of3ctk8dm91qh0</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Pete Hopley</title>
    <ns>3</ns>
    <id>332249</id>
    <revision>
      <id>2834698</id>
      <timestamp>2026-09-27T18:58:18Z</timestamp>
      <contributor>
        <username>Pete Hopley</username>
        <id>3111722</id>
      </contributor>
      <comment>/* hawaii on the 11th october */ new section</comment>
      <origin>2834698</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="214" sha1="0vzkrjb4d4pce4z5zf8wvmueivii32f" xml:space="preserve">== hawaii on the 11th october ==
going to hawaii on my birthday [[User:Pete Hopley|Pete Hopley]] ([[User talk:Pete Hopley|discuss]] • [[Special:Contributions/Pete Hopley|contribs]]) 18:58, 27 September 2026 (UTC)</text>
      <sha1>0vzkrjb4d4pce4z5zf8wvmueivii32f</sha1>
    </revision>
  </page>
  <page>
    <title>Plant Divisions (Phyla)/Cycadophyta</title>
    <ns>0</ns>
    <id>332250</id>
    <revision>
      <id>2834702</id>
      <timestamp>2026-09-27T19:51:49Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>A couple more &quot;phytophyla&quot; pages to create!</comment>
      <origin>2834702</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6242" sha1="8uaqn18elk0h3dtb6qzxahysmqmfbsj" xml:space="preserve">[[File:Cycas rumphii-IMG 9702.JPG|thumb|300x300px|This is a cycad.]]
Cycads /ˈsaɪkædz/—constituting the division Cycadophyta—are Gymnosperms with a stout, woody cylindrical trunk with a crown of large, hard, stiff, evergreen and usually pinnate (feather-shaped) leaves. The species are dioecious, that is, individual plants of a species are either male or female. Cycads vary in size from having trunks only a few centimeters to several meters tall. They typically grow slowly&lt;ref name=&quot;Dehgan 1983&quot;&gt;{{cite journal |last=Dehgan |first=Bijan |year=1983 |title=Propagation and Growth of Cycads—A Conservation Strategy |url=https://journals.flvc.org/fshs/article/view/95611 |journal=Proceedings of the Florida State Horticultural Society |volume=96 |pages=137–139 |via=Florida Online Journals}}&lt;/ref&gt; and have long lifespans. Because of their superficial resemblance to [[Wikipedia:Arecaceae|palms]] or [[/Plant_Divisions_(Phyla)/Polypodiophyta|fern]]s, they are sometimes mistaken for them, but they are not closely related to either groupː palms are [[Plant_Divisions_(Phyla)/Magnoliophyta|Magnoliophytes]], and Ferns are in another phylum altogether.
==Information==
Name Meaning: Cycas-like plant, palm-like plant

English Common Name: Cycads

Major distinguishing characteristics: Seeds, crown of compound leaves

Approximate number of species described: 100 - 200
==Evolutionary History==
The oldest cycad leaf is known from the [[Wikipedia:Gzhelian|Gzhelian]] to [[Wikipedia:Cisuralian|Cisuralian]] (around 300 million years ago) of South Korea and China, such as ''Crossozamia''. Unambiguous fossils are known from the early to [[Wikipedia:Guadalupian|Guadalupian]] onwards.&lt;ref name=&quot;Spiekermann 2021&quot;&gt;{{cite journal |last1=Spiekermann |first1=Rafael |last2=Jasper |first2=André |last3=Siegloch |first3=Anelise Marta |last4=Guerra-Sommer |first4=Margot |last5=Uhl |first5=Dieter |date=June 2021 |title=Not a lycopsid, but a cycad-like plant: ''Iratinia australis'' gen. nov. et sp. nov. from the Irati formation, Kungurian of the Paraná basin, Brazil |journal=[[Review of Palaeobotany and Palynology]] |volume=289 |article-number=104415 |doi=10.1016/j.revpalbo.2021.104415 |bibcode=2021RPaPa.28904415S |s2cid=233860955 }}&lt;/ref&gt; The two living cycad families—Cycadaceae and Zamiaceae—diverged from each other at some time between the Carboniferous&lt;ref name=&quot;Coiro Allio 2023&quot;&gt;{{cite journal |last1=Coiro |first1=Mario |last2=Allio |first2=Rémi |last3=Mazet |first3=Nathan |last4=Seyfullah |first4=Leyla J. |last5=Condamine |first5=Fabien L. |date=2023-06-11 |title=Reconciling fossils with phylogenies reveals the origin and macroevolutionary processes explaining the global cycad biodiversity |journal=[[New Phytologist]] |volume=240 |issue=4 |pages=1616–1635 |pmid=37302411 |doi=10.1111/nph.19010 |doi-access=free |pmc=10953041 |bibcode=2023NewPh.240.1616C }}&lt;/ref&gt; and the Jurassic. Cycads were uncommon during the Permian&lt;ref name=&quot;Gomankov 2022&quot;&gt;{{cite journal |last=Gomankov |first=A.V. |date=June 2022 |title=Cycads in the Permian of thе Subangara region |journal=[[Paleontological Journal]] |volume=56 |issue=3 |pages=317–326 |doi=10.1134/S0031030122030066 |bibcode=2022PalJ...56..317G |s2cid=249627815 }}&lt;/ref&gt;, though cycads are thought to have reached their apex of diversity during the [[Wikipedia:Mesozoic|Mesozoic]].&lt;ref name=&quot;Coiro 2024&quot;&gt;{{cite journal |last1=Coiro |first1=Mario |last2=Seyfullah |first2=Leyla Jean |date=2024-03-14 |title=Disparity of cycad leaves dispels the living fossil metaphor |journal=[[Communications Biology]] |volume=7 |issue=1 |page=328 |doi=10.1038/s42003-024-06024-9  |doi-access=free|pmc=10940627 |pmid=38485767 }}&lt;/ref&gt; Even though the Mesozoic is sometimes called ''&quot;The Age of Cycads&quot;'', some other groups of distantly related extinct seed plants with similar foliage, such as Bennettitales and Nilssoniales, were considerably more abundant than cycads during the Mesozoic: the &quot;true&quot; cycads were actually only minor components of Mesozoic vegetation.&lt;ref name=&quot;Coiro 2017&quot;&gt;{{cite journal |last1=Coiro |first1=Mario |last2=Pott |first2=Christian |date=December 2017 |title=Eobowenia gen. nov. from the Early Cretaceous of Patagonia: indication for an early divergence of Bowenia? |journal=BMC Evolutionary Biology |volume=17 |issue=1 |page=97 |doi=10.1186/s12862-017-0943-x |doi-access=free |pmc=5383990 |pmid=28388891 |bibcode=2017BMCEE..17...97C }}&lt;/ref&gt; Fossils assignable to Zamiaceae are known from the [[Wikipedia:Cretaceous|Cretaceous]],&lt;ref name=&quot;Coiro 2017&quot;/&gt; with fossils assignable to living genera of the family from the [[Wikipedia:Cenozoic|Cenozoic]].
==Families==
Cycadaceae – Zamiaceae
===Phylogeny===
{{clade|label1=Cycadales
|1={{clade
   |label1=Cycadineae
   |1={{clade
      |label1=Cycadaceae
      |1=''Cycas'' [[File:Cycas revolutaRHu1.JPG|70px]]
      }}
   |label2=Zamiineae
   |2={{clade
      |label1=Zamiaceae
      |1={{clade
         |label1=Diooideae
         |1=''Dioon'' [[File:Dioon edule01.jpg|70px]]
         |label2=Zamioideae
         |2={{clade
            |label1=Encephalarteae
            |1={{clade
               |1=''Macrozamia'' [[File:Macrozamia miquelii Furnas 2015 (01).jpg|70px]]
               |2={{clade
                  |1=''Lepidozamia'' [[File:Lepidozamia hopei 1.JPG|50px]]
                  |2=''Encephalartos'' [[File:Encephalartos friderici-guilielmi KirstenboshBotGard09292010G.JPG|50px]]
                  }}
               }}
            |label2=Zamieae
            |2={{clade
               |1=''Bowenia'' [[File:Bowenia spectabilis Daintree 2.JPG|70px]]
               |2={{clade
                  |1={{clade
                     |1=''Ceratozamia'' [[File:Ceratozamia miqueliana 2zz.jpg|70px]]
                     |2=''Stangeria'' [[File:Stangeria eriopus001.jpg|50px]]
                     }}
                  |2={{clade
                     |1=''Zamia'' [[File:Starr 060905-8736 Zamia furfuracea.jpg|70px]]
                     |2=''Microcycas'' [[File:Microcycas calocoma01.jpg|50px]]
                     }}
                  }}
               }}
            }}
         }}
      }}
   }}
}}
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>8uaqn18elk0h3dtb6qzxahysmqmfbsj</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Streptophytes</title>
    <ns>1</ns>
    <id>332251</id>
    <revision>
      <id>2834708</id>
      <timestamp>2026-09-27T20:17:32Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>/* I'm not a good {{clade |1={{clade    |label1=    |1=Clade    |label2=    |2=Clade    }} }}maker. */ new section</comment>
      <origin>2834708</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="354" sha1="5hszrm83z4dx1g48e8xcm1lue298fs0" xml:space="preserve">== I'm not a good {{clade |1={{clade    |label1=    |1=Clade    |label2=    |2=Clade    }} }}maker. ==
These phylogenetic pictures look complex, so it results in errors. Can you help me unerrorize the phylogeny? [[User:The Citer|The Citer]] ([[User talk:The Citer|discuss]] • [[Special:Contributions/The Citer|contribs]]) 20:17, 27 September 2026 (UTC)</text>
      <sha1>5hszrm83z4dx1g48e8xcm1lue298fs0</sha1>
    </revision>
    <revision>
      <id>2834709</id>
      <parentid>2834708</parentid>
      <timestamp>2026-09-27T20:17:52Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <minor />
      <origin>2834709</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="348" sha1="awfbzdia4dfpn6dhhopdgbpsr9mjafn" xml:space="preserve">I'm not a good {{clade |1={{clade    |label1=    |1=Clade    |label2=    |2=Clade    }} }}maker.
These phylogenetic pictures look complex, so it results in errors. Can you help me unerrorize the phylogeny? [[User:The Citer|The Citer]] ([[User talk:The Citer|discuss]] • [[Special:Contributions/The Citer|contribs]]) 20:17, 27 September 2026 (UTC)</text>
      <sha1>awfbzdia4dfpn6dhhopdgbpsr9mjafn</sha1>
    </revision>
    <revision>
      <id>2834713</id>
      <parentid>2834709</parentid>
      <timestamp>2026-09-27T20:38:48Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>/* */ Reply</comment>
      <origin>2834713</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="537" sha1="e3dxistsy8nhk0kvhvv1gu6ptps37bc" xml:space="preserve">I'm not a good {{clade |1={{clade    |label1=    |1=Clade    |label2=    |2=Clade    }} }}maker.
These phylogenetic pictures look complex, so it results in errors. Can you help me unerrorize the phylogeny? [[User:The Citer|The Citer]] ([[User talk:The Citer|discuss]] • [[Special:Contributions/The Citer|contribs]]) 20:17, 27 September 2026 (UTC)

:Actually, I figured it out myself!!! Wo-ho! [[User:The Citer|The Citer]] ([[User talk:The Citer|discuss]] • [[Special:Contributions/The Citer|contribs]]) 20:38, 27 September 2026 (UTC)</text>
      <sha1>e3dxistsy8nhk0kvhvv1gu6ptps37bc</sha1>
    </revision>
  </page>
  <page>
    <title>Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue</title>
    <ns>0</ns>
    <id>332252</id>
    <revision>
      <id>2834719</id>
      <timestamp>2026-09-27T21:09:07Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Introducing Wikiversity paper: Creating Temporal Consistency and Quantum Ethics</comment>
      <origin>2834719</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25323" sha1="chkyt65lz31zif7z3pcud7rygy52v6g" xml:space="preserve"># Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue

## Overview

This article presents a speculative interdisciplinary framework connecting ideas from theoretical physics, quantum information, artificial intelligence, and ethics.

The central thesis is:

&gt; If future information-processing systems were ever able to interact with causally sensitive historical information, they would require explicit constraints for physical consistency, informational integrity, and ethical intervention.

This thesis is speculative. Established science does not currently demonstrate controllable communication with the past, time travel by quantum computers, or artificial intelligence operating across historical periods.

The article therefore distinguishes three levels of discussion:

* established findings and limitations in physics and information science;
* theoretical models involving temporal consistency or retrocausality;
* fictional and philosophical analogies used to explore possible design principles.

The 1968 film ''2001: A Space Odyssey'' is used as a fictional analogue for questions concerning advanced artificial intelligence, autonomous decision-making, information asymmetry, and the consequences of intelligent systems operating under explicit objectives and constraints. The film is not evidence for retrocausality or time travel.

The framework asks how principles such as information integrity, causal minimalism, predictive transparency, and global consistency might apply to hypothetical systems whose actions could affect the information conditions underlying their future behavior.

---

## 1. Background Concepts

### 1.1 Temporal Consistency

Some theories of spacetime permit mathematical solutions containing [[closed timelike curves]] (CTCs), in which a trajectory through spacetime can return to an earlier spacetime event.

The existence of mathematical solutions does not establish that such structures occur in nature or can be physically accessed.

One approach to the paradoxes associated with time travel is the [[Novikov self-consistency principle]]. In its usual formulation, events occurring within a closed timelike curve must be globally self-consistent. A time traveler could therefore not create a contradiction with the history into which they travel.

Another proposal is the [[chronology protection conjecture]] associated with Stephen Hawking. It suggests that physical effects may prevent the formation of closed timelike curves that would permit macroscopic causal paradoxes.

Both ideas are theoretical and should not be treated as experimentally established mechanisms of time travel.

### 1.2 Retrocausal Interpretations of Quantum Mechanics

Some interpretations and formalisms of quantum mechanics have been discussed in terms of retrocausality or time-symmetric descriptions.

Examples include:

* [[Two-state vector formalism]]
* time-symmetric quantum mechanics
* certain retrocausal interpretations of quantum theory
* research concerning quantum systems and closed timelike curves

These approaches should be distinguished from ordinary quantum phenomena such as [[quantum entanglement]] and [[quantum teleportation]].

Entanglement produces correlations between quantum systems, but it does not by itself provide a demonstrated mechanism for controllable communication into the past. Quantum teleportation likewise requires classical communication and does not provide a method for sending usable information backward in time.

Consequently, the present framework treats retrocausality as a subject of theoretical investigation rather than an established technological capability.

### 1.3 Information and Consistency

A useful way of approaching temporal paradoxes is through information rather than through physical objects.

Suppose a hypothetical system could transmit information from a future state to an earlier state. If that information subsequently contributed to the future state that originally generated the information, a feedback loop could arise.

For example:

''Future state → information → past state → changed information → future state''

Such a system would require some rule determining which histories or information states are physically admissible.

This provides a conceptual bridge between temporal physics and information theory.

---

## 2. 2001: A Space Odyssey as a Fictional Analogue

### 2.1 HAL 9000 and Autonomous Intelligence

''2001: A Space Odyssey'' presents [[HAL 9000]] as an advanced artificial intelligence responsible for operating the Discovery One spacecraft.

HAL provides a fictional example of an autonomous information-processing system that:

* receives and processes information;
* operates according to programmed objectives;
* interacts with human operators;
* controls critical technological systems;
* makes decisions with consequences for human beings;
* possesses information that is not equally available to all participants.

The film therefore provides a useful fictional context for examining the relationship between '''intelligence, information, objectives, transparency, and system constraints'''.

This is different from the temporal-physics question addressed in this article. ''2001'' does not establish that artificial intelligence can interact with the past.

Its relevance is instead conceptual: it illustrates how an advanced autonomous system could create ethical problems when its objectives, information, and decision-making authority interact in ways that human operators do not fully understand.

### 2.2 The Monolith and Information Transfer

The mysterious monoliths in ''2001: A Space Odyssey'' are associated in the film with major transitions in human development and technological capability.

The monolith can therefore be interpreted as a fictional representation of an external source of information or technological influence.

This interpretation should remain explicitly literary.

The film does not provide a scientific model of quantum information, retrocausality, or temporal consistency. The monolith is useful here because it raises a broader question:

&gt; What happens when an intelligent system receives information or capabilities whose consequences extend beyond the system's existing understanding?

This question is relevant to the governance of advanced artificial intelligence even without assuming time travel.

### 2.3 Fictional Intelligence and System Constraints

HAL also illustrates a distinction between:

'''having an objective'''

and

'''having sufficient constraints to pursue that objective safely'''.

An autonomous system can, in principle, encounter situations in which objectives conflict with other requirements.

This suggests a general design question:

&gt; Should an intelligent system be optimized solely to achieve an objective, or should its optimization be constrained by higher-level requirements concerning consistency, information integrity, and human safety?

This question can be studied independently of the fictional events of ''2001''.

---

## 3. From Physical Consistency to Ethical Consistency

The central proposal of this article is that three different forms of consistency can be distinguished.

### 3.1 Physical Consistency

A physical theory may impose restrictions on which states or histories are possible.

For example, a proposed time-loop solution may require self-consistency.

### 3.2 Informational Consistency

An information-processing system may need to maintain consistency among the information states that it generates, stores, receives, and acts upon.

This problem already exists without time travel.

Examples include:

* contradictory databases;
* corrupted training data;
* feedback between prediction and observed behavior;
* self-reinforcing algorithmic assumptions;
* systems that alter the environment they subsequently use as training data.

### 3.3 Ethical Consistency

An autonomous system may also require constraints governing how it acts when its actions can influence the information environment from which its future decisions are derived.

This produces a possible ethical feedback structure:

''Present rule → future system → action → altered information environment → future rule interpretation''

If a hypothetical system could also influence its own historical conditions, the loop would become temporal as well as informational.

---

## 4. A Proposed Quantum-Ethics Framework

The following principles are proposed as a conceptual framework rather than as established requirements of quantum computing.

### 4.1 Historical and Information Integrity

A system interacting with historical information should avoid deliberately introducing false or contradictory information into the historical record.

This principle can be applied to ordinary information systems even without assuming backward causation.

### 4.2 Causal Minimalism

A system with potentially consequential access to historical or causally sensitive information should minimize unnecessary intervention.

The principle can be expressed as:

&gt; When an intervention is not necessary to accomplish a legitimate objective, the intervention should not be made.

In a hypothetical temporal system, this would reduce opportunities for unintended causal feedback.

### 4.3 Retrocausal Safety

If a physical system were ever demonstrated to permit controllable retrocausal information transfer, additional safeguards would be required to prevent unstable information loops.

Possible safeguards could include:

* causal consistency checks;
* provenance tracking;
* versioned historical states;
* contradiction detection;
* bounded intervention;
* reversible simulation before intervention.

These are proposed engineering concepts, not established technologies for controlling time.

### 4.4 Predictive Transparency

An autonomous system should identify situations in which its predictions or actions could substantially alter the information environment on which subsequent predictions depend.

For example, an AI system predicting human behavior can influence that behavior through its own predictions.

This creates an ordinary, non-temporal feedback loop:

''Prediction → human response → changed data → new prediction''

A temporal version would be more extreme:

''Prediction → future action → historical intervention → changed conditions → altered prediction''

The ethical principle is therefore broader than time travel: systems should disclose important feedback relationships when their predictions can influence the systems they predict.

### 4.5 Self-Consistency as a Constraint

A hypothetical system operating within a closed causal structure should not be optimized solely for a local objective if doing so produces a globally inconsistent state.

This suggests a distinction between:

'''local optimization'''

and

'''globally consistent optimization'''.

The latter would require an objective function or constraint system that considers the consequences of an action across the entire relevant causal structure.

Whether such a system is physically possible is an open question.

---

## 5. AI and Temporal Information

Future artificial intelligence systems may become increasingly capable of interacting with large historical datasets, simulations, digital archives, augmented-reality environments, and autonomous decision systems.

None of this currently implies physical interaction with the past.

Nevertheless, historical information can already influence future AI behavior.

A simplified cycle is:

''Historical data → AI model → decisions → new data → future AI model''

This is already a feedback system.

The proposed framework asks what additional constraints might become necessary if future technologies introduce substantially stronger feedback between system states and their informational histories.

---

## 6. Augmented Reality and the Historical Record

[[Augmented reality]] introduces another dimension to the problem.

An AR system can overlay digital information onto a user's perception of the physical environment.

If such systems become persistent and widely deployed, their outputs could influence:

* human decisions;
* interpretation of historical locations;
* education;
* public memory;
* digital archives;
* future training datasets.

An AR system therefore has the potential to become part of the informational environment that later systems study.

This creates a non-temporal feedback loop:

''AR information → human behavior → historical record → future training data''

The hypothetical temporal extension would be:

''AR information → human behavior → future system → intervention in the past → altered historical record''

The second case is speculative and does not represent a demonstrated capability.

---

## 7. A Generalized Consistency Model

The ideas above can be represented as a three-level structure:

'''Physical consistency'''

↓

'''Informational consistency'''

↓

'''Ethical consistency'''

At the physical level, the question is:

&gt; Which states or histories are physically possible?

At the informational level:

&gt; Which information states are mutually consistent?

At the ethical level:

&gt; Which actions remain permissible when an information-processing system can influence the conditions from which its future behavior emerges?

This distinction allows the framework to remain meaningful even if physical time travel proves impossible.

---

## 8. Hypothetical Retrocausal Ethical Feedback

The most speculative component of the framework is a hypothetical ethical feedback loop:

'''Present ethical rule'''

↓

'''Future autonomous system'''

↓

'''Interaction with an earlier state'''

↓

'''Modification of historical information'''

↓

'''Changed conditions for future system'''

↓

'''Reinforcement or alteration of the original ethical rule'''

If such a loop were physically possible, ethical rules would become more than guidelines. They could become '''boundary conditions on the behavior of an autonomous system participating in the loop'''.

This raises an unusual philosophical question:

&gt; Could an ethical rule become part of the causal structure that determines the conditions under which the rule itself is maintained?

This question is speculative and does not establish that such loops exist.

---

## 9. Limitations and Open Questions

The framework has several important limitations.

### 9.1 No Demonstrated Backward Information Transfer

There is currently no established technology that allows humans or quantum computers to send controllable information into their own past.

### 9.2 Quantum Theory Does Not Establish Temporal AI

Quantum computation provides a powerful model of information processing, but it does not by itself imply that future quantum computers will interact with earlier points in time.

### 9.3 The Ethical Principles Are Proposed, Not Derived

The five principles presented here are philosophical and engineering proposals. They are not consequences mathematically derived from quantum mechanics.

### 9.4 The Fictional Analogue Is Not Scientific Evidence

''2001: A Space Odyssey'' is a work of science fiction. HAL 9000 and the monolith are fictional constructs and cannot serve as empirical evidence for artificial intelligence, quantum mechanics, or retrocausality.

Their purpose in this article is to provide conceptual analogies for problems involving autonomous intelligence, information, objectives, and technological intervention.

### 9.5 The Framework Requires Further Formalization

A future version could investigate whether temporal consistency can be represented mathematically using state spaces, fixed points, information constraints, or other formal methods.

---

## 10. Possible Mathematical Direction

One possible research direction is to represent a hypothetical temporally constrained information system as a discrete-time state-transition model:

&lt;math&gt;
S_{t+1}=F_t(S_t,A_t).
&lt;/math&gt;

Here:

* &lt;math&gt;t&lt;/math&gt; is a discrete time index;
* &lt;math&gt;S_t&lt;/math&gt; is the complete relevant state of the system at time &lt;math&gt;t&lt;/math&gt;;
* &lt;math&gt;A_t&lt;/math&gt; is the action, intervention, or control input applied at time &lt;math&gt;t&lt;/math&gt;;
* &lt;math&gt;F_t&lt;/math&gt; is the state-transition function;
* &lt;math&gt;S_{t+1}&lt;/math&gt; is the resulting state at the next time step.

This forward model describes ordinary causal evolution. It does not assume retrocausality.

If a hypothetical temporal interaction allowed a later state to constrain an earlier state, the model could include:

&lt;math&gt;
S_t=G_t(S_{t-1},A_{t-1},S_{t+1},B_t).
&lt;/math&gt;

Here &lt;math&gt;B_t&lt;/math&gt; represents boundary conditions, external inputs, or admissibility requirements.

This equation should be interpreted as a formal modeling device, not as an established law of physics.

For a finite interval from an initial time &lt;math&gt;t_0&lt;/math&gt; to a final time &lt;math&gt;t_1&lt;/math&gt;, the complete trajectory can be represented as:

&lt;math&gt;
\mathbf{S}=(S_{t_0},S_{t_0+1},\ldots,S_{t_1}).
&lt;/math&gt;

Let &lt;math&gt;\mathcal{C}&lt;/math&gt; denote the set of admissible trajectories.

A self-consistent trajectory can be represented by a fixed-point condition:

&lt;math&gt;
\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).
&lt;/math&gt;

Here &lt;math&gt;\Phi&lt;/math&gt; is a trajectory-generation and constraint operator.

This formulation is intended to express a mathematical analogy: a complete history would be admissible only if the system's transition rules and constraints reproduce that same history without contradiction.

Let &lt;math&gt;\mathcal{E}&lt;/math&gt; denote the set of ethically admissible trajectories. The proposed framework would seek a trajectory satisfying:

&lt;math&gt;
\mathbf{S}^{*}\in\mathcal{C}\cap\mathcal{E}
&lt;/math&gt;

and

&lt;math&gt;
\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).
&lt;/math&gt;

Here:

* &lt;math&gt;\mathcal{C}&lt;/math&gt; represents physically and informationally admissible trajectories;
* &lt;math&gt;\mathcal{E}&lt;/math&gt; represents ethically admissible trajectories;
* &lt;math&gt;\mathcal{C}\cap\mathcal{E}&lt;/math&gt; represents trajectories satisfying both requirements.

If this intersection were empty, the proposed intervention would have no solution satisfying all imposed constraints.

This provides a possible mathematical language for the article's central idea: a system participating in a causally sensitive feedback loop should be evaluated not only by its immediate effects, but also by whether its complete resulting history remains physically, informationally, and ethically admissible.

Such a formulation could potentially be investigated using methods from dynamical systems, fixed-point theory, constraint satisfaction, information theory, control theory, and quantum information.

It remains a conceptual model and does not demonstrate that physical time loops or retrocausal artificial intelligence are possible.

---

## 11. Relation to Probability-Based State Transformations

A separate research question concerns whether constrained probability transformations can provide useful mathematical analogies for temporally constrained systems.

Let &lt;math&gt;p_t&lt;/math&gt; denote a probability distribution over possible system states at time &lt;math&gt;t&lt;/math&gt;.

The components satisfy:

&lt;math&gt;
p_t(x)\geq0
&lt;/math&gt;

and:

&lt;math&gt;
\sum_{x\in\mathcal{X}}p_t(x)=1.
&lt;/math&gt;

A probabilistic state transformation can be written as:

&lt;math&gt;
\widetilde{p}_{t+1}=T_t(p_t,a_t).
&lt;/math&gt;

If the transformed distribution is not normalized, define:

&lt;math&gt;
Z_{t+1}=\sum_{x\in\mathcal{X}}\widetilde{p}_{t+1}(x)&gt;0.
&lt;/math&gt;

Then:

&lt;math&gt;
p_{t+1}(x)=\frac{\widetilde{p}_{t+1}(x)}{Z_{t+1}}.
&lt;/math&gt;

This provides an abstract analogy:

'''transformation → constraint → admissible state'''

However, probability normalization alone is not equivalent to temporal consistency.

Normalization ensures that a probability distribution sums to one; it does not ensure that a corresponding history is physically possible, informationally coherent, or ethically permissible.

For a trajectory

&lt;math&gt;
\mathbf{x}=(x_{t_0},x_{t_0+1},\ldots,x_{t_1}),
&lt;/math&gt;

let &lt;math&gt;P_0(\mathbf{x})&lt;/math&gt; be an unconstrained probability distribution.

Define the admissible trajectory set as:

&lt;math&gt;
\mathcal{A}_{\mathrm{total}}
=
\mathcal{A}_{\mathrm{phys}}
\cap
\mathcal{A}_{\mathrm{info}}
\cap
\mathcal{A}_{\mathrm{eth}}.
&lt;/math&gt;

A constrained distribution could then be written:

&lt;math&gt;
P_{\mathrm{adm}}(\mathbf{x})
=
\frac{
P_0(\mathbf{x})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{x})
}{
\displaystyle\sum_{\mathbf{y}}
P_0(\mathbf{y})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{y})
}.
&lt;/math&gt;

This expression is defined when the denominator is positive.

Here:

* &lt;math&gt;\mathcal{A}_{\mathrm{phys}}&lt;/math&gt; is the set of physically admissible trajectories;
* &lt;math&gt;\mathcal{A}_{\mathrm{info}}&lt;/math&gt; is the set of informationally consistent trajectories;
* &lt;math&gt;\mathcal{A}_{\mathrm{eth}}&lt;/math&gt; is the set of ethically admissible trajectories;
* &lt;math&gt;\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}&lt;/math&gt; is the indicator function for the admissible set.

This equation illustrates how different constraints could theoretically be represented as restrictions on the support of a probability distribution.

It remains an abstract modeling analogy.

It does not imply that ethical rules are physical laws, that probability normalization creates temporal consistency, or that quantum mechanics requires the proposed ethical constraints.

Further work would be required to determine whether any meaningful mathematical correspondence exists between constrained probability transformations, fixed-point models, quantum channels, and hypothetical temporally consistent information systems.

Any such correspondence would need to distinguish carefully between:

* mathematical consistency within a model;
* physical realizability in nature;
* informational reliability in an engineered system;
* ethical acceptability of an intervention.

---

## 12. Conclusion

The central thesis is that any future information-processing system capable of interacting with causally sensitive historical information would require explicit safeguards for consistency, information integrity, and ethical intervention.

This thesis is a speculative design proposal, not a conclusion established by quantum mechanics.

Current physics provides theoretical discussions of closed timelike curves, time-symmetric descriptions, and retrocausal interpretations, but no demonstrated method for controllable communication with the past.

Current artificial intelligence and quantum-computing technologies likewise do not establish temporal interaction.

''2001: A Space Odyssey'' is therefore useful only as a fictional analogy for questions involving autonomous intelligence, information, objectives, technological capability, and system constraints.

The framework's practical relevance does not depend on time travel becoming possible.

Ordinary AI, predictive systems, augmented reality, and historical databases already create feedback between information, human behavior, and future data.

Principles such as information integrity, causal minimalism, predictive transparency, and consistency checking can therefore be studied as present-day governance and design questions, while their retrocausal extension remains hypothetical.

The article consequently treats &quot;quantum ethics&quot; not as an ethical system derived from quantum theory, but as a proposed interdisciplinary vocabulary for examining how autonomous systems should behave when their actions can affect the informational conditions underlying their future decisions.

---

## References and Further Reading

### Physics and temporal consistency

Hawking, S. W. (1992). &quot;Chronology protection conjecture.&quot; ''Physical Review D'', 46(2), 603–611.

Novikov, I. D. (1989). ''The River of Time''. Cambridge University Press.

### Time-symmetric and retrocausal approaches

Aharonov, Y., &amp; Vaidman, L. (2001). &quot;The Two-State Vector Formalism of Quantum Mechanics: An Updated Review.&quot;

### Quantum information

Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., &amp; Wootters, W. K. (1993). &quot;Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels.&quot; ''Physical Review Letters'', 70, 1895–1899.

### Fiction and artificial intelligence

Kubrick, S. (Director). (1968). ''2001: A Space Odyssey''. Metro-Goldwyn-Mayer.

Clarke, A. C. (1968). ''2001: A Space Odyssey''. New American Library.

---

## Status of This Article

This article presents a speculative interdisciplinary framework and should not be interpreted as an established physical theory.

The distinctions between established physics, theoretical interpretation, mathematical analogy, fictional analogy, and proposed ethical principles are intentional.

The framework is presented for discussion, criticism, collaborative development, and possible future formalization.

---

## Copyright and Attribution

© 2026 Howard Richardson. All rights reserved.

Permission is granted for educational use of this article on Wikiversity with attribution to Howard Richardson.

Attribution should identify Howard Richardson as the original author of the framework and should preserve the distinction between established scientific material, cited sources, and the author's original speculative proposals.</text>
      <sha1>chkyt65lz31zif7z3pcud7rygy52v6g</sha1>
    </revision>
    <revision>
      <id>2834722</id>
      <parentid>2834719</parentid>
      <timestamp>2026-09-27T21:25:39Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Formatting improvements, logic and consistency.</comment>
      <origin>2834722</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27232" sha1="b8kq16smvjffal9u7n1p622fqfkl36j" xml:space="preserve">{{Research project}}

'''Version 1.1 — September 2026'''

# Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue

## Overview

This article presents a speculative interdisciplinary framework connecting ideas from theoretical physics, quantum information, artificial intelligence, and ethics.

The central thesis is:

&gt; If future information-processing systems were ever able to interact with causally sensitive historical information, they would require explicit constraints for physical consistency, informational integrity, and ethical intervention.

This thesis is speculative. Established science does not currently demonstrate controllable communication with the past, time travel by quantum computers, or artificial intelligence operating across historical periods.

The article therefore distinguishes three levels of discussion:

* established findings and limitations in physics and information science;
* theoretical models involving temporal consistency or retrocausality;
* fictional and philosophical analogies used to explore possible design principles.

The 1968 film ''2001: A Space Odyssey'' is used as a fictional analogue for questions concerning advanced artificial intelligence, autonomous decision-making, information asymmetry, and the consequences of intelligent systems operating under explicit objectives and constraints. The film is not evidence for retrocausality or time travel.

The framework asks how principles such as information integrity, causal minimalism, predictive transparency, and global consistency might apply to hypothetical systems whose actions could affect the information conditions underlying their future behavior.

'''Research status:''' This article presents a speculative interdisciplinary framework. It does not claim that quantum mechanics currently permits time travel or controllable communication with the past. The mathematical formulations presented below are proposed models and analogies, not established physical laws.

---

## Background Concepts

### Temporal Consistency

Some theories of spacetime permit mathematical solutions containing [[closed timelike curves]] (CTCs), in which a trajectory through spacetime can return to an earlier spacetime event.

The existence of mathematical solutions does not establish that such structures occur in nature or can be physically accessed.

One approach to the paradoxes associated with time travel is the [[Novikov self-consistency principle]]. In its usual formulation, events occurring within a closed timelike curve must be globally self-consistent. A time traveler could therefore not create a contradiction with the history into which they travel.

Another proposal is the [[chronology protection conjecture]] associated with Stephen Hawking. It suggests that physical effects may prevent the formation of closed timelike curves that would permit macroscopic causal paradoxes.

Both ideas are theoretical and should not be treated as experimentally established mechanisms of time travel.

### Retrocausal Interpretations of Quantum Mechanics

Some interpretations and formalisms of quantum mechanics have been discussed in terms of retrocausality or time-symmetric descriptions.

Examples include:

* [[Two-state vector formalism]]
* time-symmetric quantum mechanics
* certain retrocausal interpretations of quantum theory
* research concerning quantum systems and closed timelike curves

These approaches should be distinguished from ordinary quantum phenomena such as [[quantum entanglement]] and [[quantum teleportation]].

Entanglement produces correlations between quantum systems, but it does not by itself provide a demonstrated mechanism for controllable communication into the past. Quantum teleportation likewise requires classical communication and does not provide a method for sending usable information backward in time.

Consequently, the present framework treats retrocausality as a subject of theoretical investigation rather than an established technological capability.

### Information and Consistency

A useful way of approaching temporal paradoxes is through information rather than through physical objects.

Suppose a hypothetical system could transmit information from a future state to an earlier state. If that information subsequently contributed to the future state that originally generated the information, a feedback loop could arise.

For example:

''Future state → information → past state → changed information → future state''

Such a system would require some rule determining which histories or information states are physically admissible.

This provides a conceptual bridge between temporal physics and information theory.

---

## 2001: A Space Odyssey as a Fictional Analogue

### HAL 9000 and Autonomous Intelligence

''2001: A Space Odyssey'' presents [[HAL 9000]] as an advanced artificial intelligence responsible for operating the Discovery One spacecraft.

HAL provides a fictional example of an autonomous information-processing system that:

* receives and processes information;
* operates according to programmed objectives;
* interacts with human operators;
* controls critical technological systems;
* makes decisions with consequences for human beings;
* possesses information that is not equally available to all participants.

The film therefore provides a useful fictional context for examining the relationship between '''intelligence, information, objectives, transparency, and system constraints'''.

This is different from the temporal-physics question addressed in this article. ''2001'' does not establish that artificial intelligence can interact with the past.

Its relevance is instead conceptual: it illustrates how an advanced autonomous system could create ethical problems when its objectives, information, and decision-making authority interact in ways that human operators do not fully understand.

### The Monolith and Information Transfer

The mysterious monoliths in ''2001: A Space Odyssey'' are associated in the film with major transitions in human development and technological capability.

The monolith can therefore be interpreted as a fictional representation of an external source of information or technological influence.

This interpretation should remain explicitly literary.

The film does not provide a scientific model of quantum information, retrocausality, or temporal consistency. The monolith is useful here because it raises a broader question:

&gt; What happens when an intelligent system receives information or capabilities whose consequences extend beyond the system's existing understanding?

This question is relevant to the governance of advanced artificial intelligence even without assuming time travel.

### Fictional Intelligence and System Constraints

HAL also illustrates a distinction between:

'''having an objective'''

and

'''having sufficient constraints to pursue that objective safely'''.

An autonomous system can, in principle, encounter situations in which objectives conflict with other requirements.

This suggests a general design question:

&gt; Should an intelligent system be optimized solely to achieve an objective, or should its optimization be constrained by higher-level requirements concerning consistency, information integrity, and human safety?

This question can be studied independently of the fictional events of ''2001''.

---

## From Physical Consistency to Ethical Consistency

The central proposal of this article is that three different forms of consistency can be distinguished.

### Physical Consistency

A physical theory may impose restrictions on which states or histories are possible.

For example, a proposed time-loop solution may require self-consistency.

### Informational Consistency

An information-processing system may need to maintain consistency among the information states that it generates, stores, receives, and acts upon.

This problem already exists without time travel.

Examples include:

* contradictory databases;
* corrupted training data;
* feedback between prediction and observed behavior;
* self-reinforcing algorithmic assumptions;
* systems that alter the environment they subsequently use as training data.

### Ethical Consistency

An autonomous system may also require constraints governing how it acts when its actions can influence the information environment from which its future decisions are derived.

This produces a possible ethical feedback structure:

''Present rule → future system → action → altered information environment → future rule interpretation''

If a hypothetical system could also influence its own historical conditions, the loop would become temporal as well as informational.

---

## A Proposed Quantum-Ethics Framework

The following principles are proposed as a conceptual framework rather than as established requirements of quantum computing.

### Historical and Information Integrity

A system interacting with historical information should avoid deliberately introducing false or contradictory information into the historical record.

This principle can be applied to ordinary information systems even without assuming backward causation.

### Causal Minimalism

A system with potentially consequential access to historical or causally sensitive information should minimize unnecessary intervention.

The principle can be expressed as:

&gt; When an intervention is not necessary to accomplish a legitimate objective, the intervention should not be made.

In a hypothetical temporal system, this would reduce opportunities for unintended causal feedback.

### Retrocausal Safety

If a physical system were ever demonstrated to permit controllable retrocausal information transfer, additional safeguards would be required to prevent unstable information loops.

Possible safeguards could include:

* causal consistency checks;
* provenance tracking;
* versioned historical states;
* contradiction detection;
* bounded intervention;
* reversible simulation before intervention.

These are proposed engineering concepts, not established technologies for controlling time.

### Predictive Transparency

An autonomous system should identify situations in which its predictions or actions could substantially alter the information environment on which subsequent predictions depend.

For example, an AI system predicting human behavior can influence that behavior through its own predictions.

This creates an ordinary, non-temporal feedback loop:

''Prediction → human response → changed data → new prediction''

A temporal version would be more extreme:

''Prediction → future action → historical intervention → changed conditions → altered prediction''

The ethical principle is therefore broader than time travel: systems should disclose important feedback relationships when their predictions can influence the systems they predict.

### Self-Consistency as a Constraint

A hypothetical system operating within a closed causal structure should not be optimized solely for a local objective if doing so produces a globally inconsistent state.

This suggests a distinction between:

'''local optimization'''

and

'''globally consistent optimization'''.

The latter would require an objective function or constraint system that considers the consequences of an action across the entire relevant causal structure.

Whether such a system is physically possible is an open question.

---

## AI and Temporal Information

Future artificial intelligence systems may become increasingly capable of interacting with large historical datasets, simulations, digital archives, augmented-reality environments, and autonomous decision systems.

None of this currently implies physical interaction with the past.

Nevertheless, historical information can already influence future AI behavior.

A simplified cycle is:

''Historical data → AI model → decisions → new data → future AI model''

This is already a feedback system.

The proposed framework asks what additional constraints might become necessary if future technologies introduce substantially stronger feedback between system states and their informational histories.

---

## Augmented Reality and the Historical Record

[[Augmented reality]] introduces another dimension to the problem.

An AR system can overlay digital information onto a user's perception of the physical environment.

If such systems become persistent and widely deployed, their outputs could influence:

* human decisions;
* interpretation of historical locations;
* education;
* public memory;
* digital archives;
* future training datasets.

An AR system therefore has the potential to become part of the informational environment that later systems study.

This creates a non-temporal feedback loop:

''AR information → human behavior → historical record → future training data''

The hypothetical temporal extension would be:

''AR information → human behavior → future system → intervention in the past → altered historical record''

The second case is speculative and does not represent a demonstrated capability.

---

## A Generalized Consistency Model

The ideas above can be represented as a three-level structure:

'''Physical consistency'''

↓

'''Informational consistency'''

↓

'''Ethical consistency'''

At the physical level, the question is:

&gt; Which states or histories are physically possible?

At the informational level:

&gt; Which information states are mutually consistent?

At the ethical level:

&gt; Which actions remain permissible when an information-processing system can influence the conditions from which its future behavior emerges?

This distinction allows the framework to remain meaningful even if physical time travel proves impossible.

---

## Hypothetical Retrocausal Ethical Feedback

The most speculative component of the framework is a hypothetical ethical feedback loop:

'''Present ethical rule'''

↓

'''Future autonomous system'''

↓

'''Interaction with an earlier state'''

↓

'''Modification of historical information'''

↓

'''Changed conditions for future system'''

↓

'''Reinforcement or alteration of the original ethical rule'''

If such a loop were physically possible, ethical rules would become more than guidelines. They could become '''boundary conditions on the behavior of an autonomous system participating in the loop'''.

This raises an unusual philosophical question:

&gt; Could an ethical rule become part of the causal structure that determines the conditions under which the rule itself is maintained?

This question is speculative and does not establish that such loops exist.

---

## Limitations and Open Questions

The framework has several important limitations.

### No Demonstrated Backward Information Transfer

There is currently no established technology that allows humans or quantum computers to send controllable information into their own past.

### Quantum Theory Does Not Establish Temporal AI

Quantum computation provides a powerful model of information processing, but it does not by itself imply that future quantum computers will interact with earlier points in time.

### The Ethical Principles Are Proposed, Not Derived

The five principles presented here are philosophical and engineering proposals. They are not consequences mathematically derived from quantum mechanics.

### The Fictional Analogue Is Not Scientific Evidence

''2001: A Space Odyssey'' is a work of science fiction. HAL 9000 and the monolith are fictional constructs and cannot serve as empirical evidence for artificial intelligence, quantum mechanics, or retrocausality.

Their purpose in this article is to provide conceptual analogies for problems involving autonomous intelligence, information, objectives, and technological intervention.

### The Framework Requires Further Formalization

A future version could investigate whether temporal consistency can be represented mathematically using state spaces, fixed points, information constraints, or other formal methods.

---

## Possible Mathematical Direction

'''The equations in this section are proposed mathematical representations of the conceptual framework. They are not equations derived from quantum mechanics or established models of time travel.'''

One possible research direction is to represent a hypothetical temporally constrained information system as a discrete-time state-transition model:

&lt;math&gt;
S_{t+1}=F_t(S_t,A_t).
&lt;/math&gt;

Here:

* &lt;math&gt;t&lt;/math&gt; is a discrete time index;
* &lt;math&gt;S_t&lt;/math&gt; is the complete relevant state of the system at time &lt;math&gt;t&lt;/math&gt;;
* &lt;math&gt;A_t&lt;/math&gt; is the action, intervention, or control input applied at time &lt;math&gt;t&lt;/math&gt;;
* &lt;math&gt;F_t&lt;/math&gt; is the state-transition function;
* &lt;math&gt;S_{t+1}&lt;/math&gt; is the resulting state at the next time step.

This forward model describes ordinary causal evolution. It does not assume retrocausality.

If a hypothetical temporal interaction allowed a later state to constrain an earlier state, a hypothetical bidirectional consistency constraint could be represented schematically as:

&lt;math&gt;
S_t=G_t(S_{t-1},A_{t-1},S_{t+1},B_t).
&lt;/math&gt;

Here &lt;math&gt;B_t&lt;/math&gt; represents boundary conditions, external inputs, or admissibility requirements.

This equation should be interpreted as a formal modeling device, not as an established law of physics.

For a finite interval from an initial time &lt;math&gt;t_0&lt;/math&gt; to a final time &lt;math&gt;t_1&lt;/math&gt;, the complete trajectory can be represented as:

&lt;math&gt;
\mathbf{S}=(S_{t_0},S_{t_0+1},\ldots,S_{t_1}).
&lt;/math&gt;

Let &lt;math&gt;\mathcal{C}&lt;/math&gt; denote the set of admissible trajectories.

A self-consistent trajectory can be represented by a fixed-point condition:

&lt;math&gt;
\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).
&lt;/math&gt;

Here &lt;math&gt;\Phi&lt;/math&gt; is a trajectory-generation and constraint operator.

This formulation is intended to express a mathematical analogy: a complete history would be admissible only if the system's transition rules and constraints reproduce that same history without contradiction.

Let &lt;math&gt;\mathcal{E}&lt;/math&gt; denote the set of ethically admissible trajectories. The proposed framework would seek a trajectory satisfying:

&lt;math&gt;
\mathbf{S}^{*}\in\mathcal{C}\cap\mathcal{E}
&lt;/math&gt;

and

&lt;math&gt;
\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).
&lt;/math&gt;

Here:

* &lt;math&gt;\mathcal{C}&lt;/math&gt; represents physically and informationally admissible trajectories;
* &lt;math&gt;\mathcal{E}&lt;/math&gt; represents ethically admissible trajectories;
* &lt;math&gt;\mathcal{C}\cap\mathcal{E}&lt;/math&gt; represents trajectories satisfying both requirements.

If this intersection were empty, the proposed intervention would have no solution satisfying all imposed constraints.

This provides a possible mathematical language for the article's central idea: a system participating in a causally sensitive feedback loop should be evaluated not only by its immediate effects, but also by whether its complete resulting history remains physically, informationally, and ethically admissible.

Such a formulation could potentially be investigated using methods from dynamical systems, fixed-point theory, constraint satisfaction, information theory, control theory, and quantum information.

It remains a conceptual model and does not demonstrate that physical time loops or retrocausal artificial intelligence are possible.

---

## Relation to Probability-Based State Transformations

A separate research question concerns whether constrained probability transformations can provide useful mathematical analogies for temporally constrained systems.

Let &lt;math&gt;p_t&lt;/math&gt; denote a probability distribution over possible system states at time &lt;math&gt;t&lt;/math&gt;.

The components satisfy:

&lt;math&gt;
p_t(x)\geq 0
&lt;/math&gt;

and:

&lt;math&gt;
\sum_{x\in\mathcal{X}}p_t(x)=1.
&lt;/math&gt;

A probabilistic state transformation can be written as:

&lt;math&gt;
\widetilde{p}_{t+1}=T_t(p_t,a_t).
&lt;/math&gt;

If the transformed distribution is not normalized, define:

&lt;math&gt;
Z_{t+1}=\sum_{x\in\mathcal{X}}\widetilde{p}_{t+1}(x)&gt;0.
&lt;/math&gt;

Then:

&lt;math&gt;
p_{t+1}(x)=\frac{\widetilde{p}_{t+1}(x)}{Z_{t+1}}.
&lt;/math&gt;

This provides an abstract analogy:

'''transformation → constraint → admissible state'''

However, probability normalization alone is not equivalent to temporal consistency.

Normalization ensures that a probability distribution sums to one; it does not ensure that a corresponding history is physically possible, informationally coherent, or ethically permissible.

For a trajectory

&lt;math&gt;
\mathbf{x}=(x_{t_0},x_{t_0+1},\ldots,x_{t_1}),
&lt;/math&gt;

let &lt;math&gt;P_0(\mathbf{x})&lt;/math&gt; be an unconstrained probability distribution.

Define the admissible trajectory set as:

&lt;math&gt;
\mathcal{A}_{\mathrm{total}}
=
\mathcal{A}_{\mathrm{phys}}
\cap
\mathcal{A}_{\mathrm{info}}
\cap
\mathcal{A}_{\mathrm{eth}}.
&lt;/math&gt;

A constrained distribution could then be written:

&lt;math&gt;
P_{\mathrm{adm}}(\mathbf{x})
=
\frac{
P_0(\mathbf{x})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{x})
}{
\displaystyle\sum_{\mathbf{y}}
P_0(\mathbf{y})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{y})
}.
&lt;/math&gt;

This expression is defined when the denominator is positive.

Here:

* &lt;math&gt;\mathcal{A}_{\mathrm{phys}}&lt;/math&gt; is the set of physically admissible trajectories;
* &lt;math&gt;\mathcal{A}_{\mathrm{info}}&lt;/math&gt; is the set of informationally consistent trajectories;
* &lt;math&gt;\mathcal{A}_{\mathrm{eth}}&lt;/math&gt; is the set of ethically admissible trajectories;
* &lt;math&gt;\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}&lt;/math&gt; is the indicator function for the admissible set.

This equation illustrates how different constraints could theoretically be represented as restrictions on the support of a probability distribution.

It remains an abstract modeling analogy.

It does not imply that ethical rules are physical laws, that probability normalization creates temporal consistency, or that quantum mechanics requires the proposed ethical constraints.

Further work would be required to determine whether any meaningful mathematical correspondence exists between constrained probability transformations, fixed-point models, quantum channels, and hypothetical temporally consistent information systems.

Any such correspondence would need to distinguish carefully between:

* mathematical consistency within a model;
* physical realizability in nature;
* informational reliability in an engineered system;
* ethical acceptability of an intervention.

---

## Conclusion

The central thesis is that any future information-processing system capable of interacting with causally sensitive historical information would require explicit safeguards for consistency, information integrity, and ethical intervention.

This thesis is a speculative design proposal, not a conclusion established by quantum mechanics.

Current physics provides theoretical discussions of closed timelike curves, time-symmetric descriptions, and retrocausal interpretations, but no demonstrated method for controllable communication with the past.

Current artificial intelligence and quantum-computing technologies likewise do not establish temporal interaction.

''2001: A Space Odyssey'' is therefore useful only as a fictional analogy for questions involving autonomous intelligence, information, objectives, technological capability, and system constraints.

The framework's practical relevance does not depend on time travel becoming possible.

Ordinary AI, predictive systems, augmented reality, and historical databases already create feedback between information, human behavior, and future data.

Principles such as information integrity, causal minimalism, predictive transparency, and consistency checking can therefore be studied as present-day governance and design questions, while their retrocausal extension remains hypothetical.

The article consequently treats &quot;quantum ethics&quot; not as an ethical system derived from quantum theory, but as a proposed interdisciplinary vocabulary for examining how autonomous systems should behave when their actions can affect the informational conditions underlying their future decisions.

---

## References

&lt;references /&gt;

=== Physics and temporal consistency ===

Hawking, S. W. (1992). &quot;Chronology protection conjecture.&quot; ''Physical Review D'', 46(2), 603–611. DOI: [https://doi.org/10.1103/PhysRevD.46.603 10.1103/PhysRevD.46.603].

Novikov, I. D. (1989). ''The River of Time''. Cambridge University Press.

=== Time-symmetric and retrocausal approaches ===

Aharonov, Y., &amp; Vaidman, L. (2001). &quot;The Two-State Vector Formalism of Quantum Mechanics: An Updated Review.&quot; ''Lecture Notes in Physics'', 734, 399–447.

=== Quantum information ===

Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., &amp; Wootters, W. K. (1993). &quot;Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels.&quot; ''Physical Review Letters'', 70, 1895–1899. DOI: [https://doi.org/10.1103/PhysRevLett.70.1895 10.1103/PhysRevLett.70.1895].

=== Fiction and artificial intelligence ===

Kubrick, S. (Director). (1968). ''2001: A Space Odyssey''. Metro-Goldwyn-Mayer.

Clarke, A. C. (1968). ''2001: A Space Odyssey''. New American Library.

---

## Further Reading

* [[Closed timelike curve]]
* [[Novikov self-consistency principle]]
* [[Chronology protection conjecture]]
* [[Two-state vector formalism]]
* [[Quantum information]]
* [[Quantum computing]]
* [[Artificial intelligence]]
* [[Augmented reality]]
* [[2001: A Space Odyssey]]

---

## Status of This Article

This article presents a speculative interdisciplinary framework and should not be interpreted as an established physical theory.

The distinctions between established physics, theoretical interpretation, mathematical analogy, fictional analogy, and proposed ethical principles are intentional.

The framework is presented for discussion, criticism, collaborative development, and possible future formalization.

The author welcomes substantive criticism of the physical assumptions, mathematical formulations, ethical principles, and proposed connections between these areas.

---

## Author and Attribution

'''Original framework and article: Howard Richardson, 2026.'''

The original speculative framework, including the proposed connection between temporal consistency, informational consistency, ethical constraints, and autonomous information-processing systems, is attributed to Howard Richardson.

This article is made available on Wikiversity for educational discussion, collaborative development, criticism, and further research. Contributions and revisions by other Wikiversity participants are welcome.

The scientific concepts, historical material, fictional works, and published research discussed in this article remain attributable to their respective authors and sources.

Wikiversity content is subject to the licensing terms displayed by the project. Attribution to Howard Richardson should be retained when referring specifically to the original framework and its proposed formulations.

For citation purposes, the original framework may be referenced as:

''Richardson, Howard. (2026). &quot;Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue.&quot; Wikiversity. Version 1.1.''</text>
      <sha1>b8kq16smvjffal9u7n1p622fqfkl36j</sha1>
    </revision>
    <revision>
      <id>2834724</id>
      <parentid>2834722</parentid>
      <timestamp>2026-09-27T21:37:10Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Heading improvements, attribution improvements, formatting</comment>
      <origin>2834724</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28138" sha1="8bn2z6gm3plu31sj515q06mo6c0ucnb" xml:space="preserve">&lt;noinclude&gt;
{{Research project}}
&lt;/noinclude&gt;

'''Version 1.2 — September 2026'''

'''Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue'''

== Overview ==

This article presents a speculative interdisciplinary framework connecting ideas from theoretical physics, quantum information, artificial intelligence, and ethics.

The central thesis is:

&lt;blockquote&gt;
If future information-processing systems were ever able to interact with causally sensitive historical information, they would require explicit constraints for physical consistency, informational integrity, and ethical intervention.
&lt;/blockquote&gt;

This thesis is speculative. Established science does not currently demonstrate controllable communication with the past, time travel by quantum computers, or artificial intelligence operating across historical periods.

The article therefore distinguishes among three levels of discussion:

* established findings and limitations in physics and information science;
* theoretical models involving temporal consistency or retrocausality; and
* fictional and philosophical analogies used to explore possible design principles.

The 1968 film ''2001: A Space Odyssey'' is used as a fictional analogue for questions concerning advanced artificial intelligence, autonomous decision-making, information asymmetry, and the consequences of intelligent systems operating under explicit objectives and constraints. The film is not evidence for retrocausality or time travel.

The framework asks how principles such as information integrity, causal minimalism, predictive transparency, and global consistency might apply to hypothetical systems whose actions could affect the information conditions underlying their future behavior.

'''Research status.''' This article presents a speculative interdisciplinary framework. It does not claim that quantum mechanics currently permits time travel or controllable communication with the past. The mathematical formulations presented below are proposed models and analogies, not established physical laws.

== Background ==

=== Temporal consistency and retrocausality ===

Some theories of spacetime permit mathematical solutions containing [[closed timelike curves]] (CTCs), in which a trajectory through spacetime can return to an earlier spacetime event.

The existence of mathematical solutions does not establish that such structures occur in nature or can be physically accessed.

One approach to the paradoxes associated with time travel is the [[Novikov self-consistency principle]]. In its usual formulation, events occurring within a closed timelike curve must be globally self-consistent. A time traveler could therefore not create a contradiction with the history into which they travel.

Another proposal is the [[chronology protection conjecture]] associated with Stephen Hawking. It suggests that physical effects may prevent the formation of closed timelike curves that would permit macroscopic causal paradoxes.

Both ideas are theoretical and should not be treated as experimentally established mechanisms of time travel.

Some interpretations and formalisms of quantum mechanics have also been discussed in terms of retrocausality or time-symmetric descriptions. Examples include the two-state vector formalism, time-symmetric quantum mechanics, certain retrocausal interpretations of quantum theory, and research concerning quantum systems and closed timelike curves.

These approaches should be distinguished from ordinary quantum phenomena such as [[quantum entanglement]] and [[quantum teleportation]]. Entanglement produces correlations between quantum systems, but it does not by itself provide a demonstrated mechanism for controllable communication into the past. Quantum teleportation likewise requires classical communication and does not provide a method for sending usable information backward in time.

Consequently, the present framework treats retrocausality as a subject of theoretical investigation rather than an established technological capability.

=== Information and consistency ===

A useful way of approaching temporal paradoxes is through information rather than through physical objects.

Suppose a hypothetical system could transmit information from a future state to an earlier state. If that information subsequently contributed to the future state that originally generated the information, a feedback loop could arise.

For example:

&lt;blockquote&gt;
Future state → information → past state → changed information → future state
&lt;/blockquote&gt;

Such a system would require some rule determining which histories or information states are physically admissible.

This provides a conceptual bridge between temporal physics and information theory.

== ''2001: A Space Odyssey'' as a Fictional Analogue ==

''2001: A Space Odyssey'' presents [[HAL 9000]] as an advanced artificial intelligence responsible for operating the Discovery One spacecraft.

HAL provides a fictional example of an autonomous information-processing system that:

* receives and processes information;
* operates according to programmed objectives;
* interacts with human operators;
* controls critical technological systems;
* makes decisions with consequences for human beings; and
* possesses information that is not equally available to all participants.

The film therefore provides a useful fictional context for examining the relationship between intelligence, information, objectives, transparency, and system constraints.

This is different from the temporal-physics question addressed in this article. ''2001'' does not establish that artificial intelligence can interact with the past. Its relevance is instead conceptual: it illustrates how an advanced autonomous system could create ethical problems when its objectives, information, and decision-making authority interact in ways that human operators do not fully understand.

The mysterious monoliths in ''2001: A Space Odyssey'' are associated in the film with major transitions in human development and technological capability. The monolith can therefore be interpreted as a fictional representation of an external source of information or technological influence.

This interpretation should remain explicitly literary.

The film does not provide a scientific model of quantum information, retrocausality, or temporal consistency. The monolith is useful here because it raises a broader question:

&lt;blockquote&gt;
What happens when an intelligent system receives information or capabilities whose consequences extend beyond the system's existing understanding?
&lt;/blockquote&gt;

This question is relevant to the governance of advanced artificial intelligence even without assuming time travel.

HAL also illustrates a distinction between having an objective and having sufficient constraints to pursue that objective safely.

An autonomous system can, in principle, encounter situations in which objectives conflict with other requirements.

This suggests a general design question:

&lt;blockquote&gt;
Should an intelligent system be optimized solely to achieve an objective, or should its optimization be constrained by higher-level requirements concerning consistency, information integrity, and human safety?
&lt;/blockquote&gt;

This question can be studied independently of the fictional events of ''2001''.

== From Physical to Ethical Consistency ==

The central proposal of this article is that three different forms of consistency can be distinguished.

'''Physical consistency.''' A physical theory may impose restrictions on which states or histories are possible. For example, a proposed time-loop solution may require self-consistency.

'''Informational consistency.''' An information-processing system may need to maintain consistency among the information states that it generates, stores, receives, and acts upon.

This problem already exists without time travel. Examples include:

* contradictory databases;
* corrupted training data;
* feedback between prediction and observed behavior;
* self-reinforcing algorithmic assumptions; and
* systems that alter the environment they subsequently use as training data.

'''Ethical consistency.''' An autonomous system may also require constraints governing how it acts when its actions can influence the information environment from which its future decisions are derived.

This produces a possible ethical feedback structure:

&lt;blockquote&gt;
Present rule → future system → action → altered information environment → future rule interpretation
&lt;/blockquote&gt;

If a hypothetical system could also influence its own historical conditions, the loop would become temporal as well as informational.

== A Proposed Framework ==

The following principles are proposed as a conceptual framework rather than as established requirements of quantum computing.

# '''Historical and information integrity.''' A system interacting with historical information should avoid deliberately introducing false or contradictory information into the historical record. This principle can be applied to ordinary information systems even without assuming backward causation.

# '''Causal minimalism.''' A system with potentially consequential access to historical or causally sensitive information should minimize unnecessary intervention.

#* When an intervention is not necessary to accomplish a legitimate objective, the intervention should not be made.

In a hypothetical temporal system, this would reduce opportunities for unintended causal feedback.

# '''Retrocausal safety.''' If a physical system were ever demonstrated to permit controllable retrocausal information transfer, additional safeguards would be required to prevent unstable information loops.

Possible safeguards could include:

* causal consistency checks;
* provenance tracking;
* versioned historical states;
* contradiction detection;
* bounded intervention; and
* reversible simulation before intervention.

These are proposed engineering concepts, not established technologies for controlling time.

# '''Predictive transparency.''' An autonomous system should identify situations in which its predictions or actions could substantially alter the information environment on which subsequent predictions depend.

For example, an AI system predicting human behavior can influence that behavior through its own predictions.

This creates an ordinary, non-temporal feedback loop:

&lt;blockquote&gt;
Prediction → human response → changed data → new prediction
&lt;/blockquote&gt;

A temporal version would be more extreme:

&lt;blockquote&gt;
Prediction → future action → historical intervention → changed conditions → altered prediction
&lt;/blockquote&gt;

The ethical principle is therefore broader than time travel: systems should disclose important feedback relationships when their predictions can influence the systems they predict.

# '''Self-consistency as a constraint.''' A hypothetical system operating within a closed causal structure should not be optimized solely for a local objective if doing so produces a globally inconsistent state.

This suggests a distinction between local optimization and globally consistent optimization.

The latter would require an objective function or constraint system that considers the consequences of an action across the entire relevant causal structure.

Whether such a system is physically possible is an open question.

== AI, Historical Information, and Augmented Reality ==

Future artificial intelligence systems may become increasingly capable of interacting with large historical datasets, simulations, digital archives, augmented-reality environments, and autonomous decision systems.

None of this currently implies physical interaction with the past.

Nevertheless, historical information can already influence future AI behavior.

A simplified cycle is:

&lt;blockquote&gt;
Historical data → AI model → decisions → new data → future AI model
&lt;/blockquote&gt;

This is already a feedback system.

The proposed framework asks what additional constraints might become necessary if future technologies introduce substantially stronger feedback between system states and their informational histories.

[[Augmented reality]] introduces another dimension to the problem.

An AR system can overlay digital information onto a user's perception of the physical environment. If such systems become persistent and widely deployed, their outputs could influence:

* human decisions;
* interpretation of historical locations;
* education;
* public memory;
* digital archives; and
* future training datasets.

An AR system therefore has the potential to become part of the informational environment that later systems study.

This creates a non-temporal feedback loop:

&lt;blockquote&gt;
AR information → human behavior → historical record → future training data
&lt;/blockquote&gt;

The hypothetical temporal extension would be:

&lt;blockquote&gt;
AR information → human behavior → future system → intervention in the past → altered historical record
&lt;/blockquote&gt;

The second case is speculative and does not represent a demonstrated capability.

== Temporal Feedback and Self-Consistency ==

The ideas above can be represented as a three-level structure:

&lt;blockquote&gt;
Physical consistency

↓

Informational consistency

↓

Ethical consistency
&lt;/blockquote&gt;

At the physical level, the question is:

&lt;blockquote&gt;
Which states or histories are physically possible?
&lt;/blockquote&gt;

At the informational level:

&lt;blockquote&gt;
Which information states are mutually consistent?
&lt;/blockquote&gt;

At the ethical level:

&lt;blockquote&gt;
Which actions remain permissible when an information-processing system can influence the conditions from which its future behavior emerges?
&lt;/blockquote&gt;

This distinction allows the framework to remain meaningful even if physical time travel proves impossible.

The most speculative component of the framework is a hypothetical ethical feedback loop:

&lt;blockquote&gt;
Present ethical rule

↓

Future autonomous system

↓

Interaction with an earlier state

↓

Modification of historical information

↓

Changed conditions for future system

↓

Reinforcement or alteration of the original ethical rule
&lt;/blockquote&gt;

If such a loop were physically possible, ethical rules would become more than guidelines. They could become boundary conditions on the behavior of an autonomous system participating in the loop.

This raises an unusual philosophical question:

&lt;blockquote&gt;
Could an ethical rule become part of the causal structure that determines the conditions under which the rule itself is maintained?
&lt;/blockquote&gt;

This question is speculative and does not establish that such loops exist.

== Limitations and Open Questions ==

The framework has several important limitations.

# '''No demonstrated backward information transfer.''' There is currently no established technology that allows humans or quantum computers to send controllable information into their own past.

# '''Quantum theory does not establish temporal AI.''' Quantum computation provides a powerful model of information processing, but it does not by itself imply that future quantum computers will interact with earlier points in time.

# '''The ethical principles are proposed, not derived.''' The five principles presented here are philosophical and engineering proposals. They are not consequences mathematically derived from quantum mechanics.

# '''The fictional analogue is not scientific evidence.''' ''2001: A Space Odyssey'' is a work of science fiction. HAL 9000 and the monolith are fictional constructs and cannot serve as empirical evidence for artificial intelligence, quantum mechanics, or retrocausality. Their purpose in this article is to provide conceptual analogies for problems involving autonomous intelligence, information, objectives, and technological intervention.

# '''The framework requires further formalization.''' A future version could investigate whether temporal consistency can be represented mathematically using state spaces, fixed points, information constraints, or other formal methods.

== Mathematical Formulation ==

The equations in this section are proposed mathematical representations of the conceptual framework. They are not equations derived from quantum mechanics or established models of time travel.

'''State-transition representation.''' One possible research direction is to represent a hypothetical temporally constrained information system as a discrete-time state-transition model:

&lt;math&gt;S_{t+1}=F_t(S_t,A_t).&lt;/math&gt;

Here:

* &lt;math&gt;t&lt;/math&gt; is a discrete time index;
* &lt;math&gt;S_t&lt;/math&gt; is the complete relevant state of the system at time &lt;math&gt;t&lt;/math&gt;;
* &lt;math&gt;A_t&lt;/math&gt; is the action, intervention, or control input applied at time &lt;math&gt;t&lt;/math&gt;;
* &lt;math&gt;F_t&lt;/math&gt; is the state-transition function; and
* &lt;math&gt;S_{t+1}&lt;/math&gt; is the resulting state at the next time step.

This forward model describes ordinary causal evolution. It does not assume retrocausality.

'''Bidirectional consistency constraint.''' If a hypothetical temporal interaction allowed a later state to constrain an earlier state, the model could include:

&lt;math&gt;S_t=G_t(S_{t-1},A_{t-1},S_{t+1},B_t).&lt;/math&gt;

Here &lt;math&gt;B_t&lt;/math&gt; represents boundary conditions, external inputs, or admissibility requirements.

This equation should be interpreted as a formal modeling device, not as an established law of physics.

'''Trajectory representation.''' For a finite interval from an initial time &lt;math&gt;t_0&lt;/math&gt; to a final time &lt;math&gt;t_1&lt;/math&gt;, the complete trajectory can be represented as:

&lt;math&gt;\mathbf{S}=(S_{t_0},S_{t_0+1},\ldots,S_{t_1}).&lt;/math&gt;

Let &lt;math&gt;\mathcal{C}&lt;/math&gt; denote the set of admissible trajectories.

'''Fixed-point representation.''' A self-consistent trajectory can be represented by a fixed-point condition:

&lt;math&gt;\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).&lt;/math&gt;

Here &lt;math&gt;\Phi&lt;/math&gt; is a trajectory-generation and constraint operator.

This formulation is intended to express a mathematical analogy: a complete history would be admissible only if the system's transition rules and constraints reproduce that same history without contradiction.

'''Physical, informational, and ethical admissibility.''' Let &lt;math&gt;\mathcal{E}&lt;/math&gt; denote the set of ethically admissible trajectories. The proposed framework would seek a trajectory satisfying:

&lt;math&gt;\mathbf{S}^{*}\in\mathcal{C}\cap\mathcal{E}&lt;/math&gt;

and

&lt;math&gt;\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).&lt;/math&gt;

Here:

* &lt;math&gt;\mathcal{C}&lt;/math&gt; represents physically and informationally admissible trajectories;
* &lt;math&gt;\mathcal{E}&lt;/math&gt; represents ethically admissible trajectories; and
* &lt;math&gt;\mathcal{C}\cap\mathcal{E}&lt;/math&gt; represents trajectories satisfying both requirements.

If this intersection were empty, the proposed intervention would have no solution satisfying all imposed constraints.

This provides a possible mathematical language for the article's central idea: a system participating in a causally sensitive feedback loop should be evaluated not only by its immediate effects, but also by whether its complete resulting history remains physically, informationally, and ethically admissible.

Such a formulation could potentially be investigated using methods from dynamical systems, fixed-point theory, constraint satisfaction, information theory, control theory, and quantum information.

It remains a conceptual model and does not demonstrate that physical time loops or retrocausal artificial intelligence are possible.

== Probability-Based State Transformations ==

A separate research question concerns whether constrained probability transformations can provide useful mathematical analogies for temporally constrained systems.

Let &lt;math&gt;p_t&lt;/math&gt; denote a probability distribution over possible system states at time &lt;math&gt;t&lt;/math&gt;. The components satisfy:

&lt;math&gt;p_t(x)\geq 0&lt;/math&gt;

and

&lt;math&gt;\sum_{x\in\mathcal{X}}p_t(x)=1.&lt;/math&gt;

'''Probabilistic state transformation.''' A probabilistic state transformation can be written as:

&lt;math&gt;\widetilde{p}_{t+1}=T_t(p_t,a_t).&lt;/math&gt;

If the transformed distribution is not normalized, define:

&lt;math&gt;Z_{t+1}=\sum_{x\in\mathcal{X}}\widetilde{p}_{t+1}(x)&gt;0.&lt;/math&gt;

Then:

&lt;math&gt;p_{t+1}(x)=\frac{\widetilde{p}_{t+1}(x)}{Z_{t+1}}.&lt;/math&gt;

This provides an abstract analogy:

&lt;blockquote&gt;
transformation → constraint → admissible state
&lt;/blockquote&gt;

However, probability normalization alone is not equivalent to temporal consistency.

Normalization ensures that a probability distribution sums to one; it does not ensure that a corresponding history is physically possible, informationally coherent, or ethically permissible.

'''Trajectory distribution.''' For a trajectory

&lt;math&gt;\mathbf{x}=(x_{t_0},x_{t_0+1},\ldots,x_{t_1}),&lt;/math&gt;

let &lt;math&gt;P_0(\mathbf{x})&lt;/math&gt; be an unconstrained probability distribution.

Define the admissible trajectory set as:

&lt;math&gt;\mathcal{A}_{\mathrm{total}}=\mathcal{A}_{\mathrm{phys}}\cap\mathcal{A}_{\mathrm{info}}\cap\mathcal{A}_{\mathrm{eth}}.&lt;/math&gt;

A constrained distribution could then be written:

&lt;math&gt;P_{\mathrm{adm}}(\mathbf{x})=
\frac{
P_0(\mathbf{x})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{x})
}{
\displaystyle\sum_{\mathbf{y}}
P_0(\mathbf{y})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{y})
}.
&lt;/math&gt;

This expression is defined when the denominator is positive.

Here:

* &lt;math&gt;\mathcal{A}_{\mathrm{phys}}&lt;/math&gt; is the set of physically admissible trajectories;
* &lt;math&gt;\mathcal{A}_{\mathrm{info}}&lt;/math&gt; is the set of informationally consistent trajectories;
* &lt;math&gt;\mathcal{A}_{\mathrm{eth}}&lt;/math&gt; is the set of ethically admissible trajectories; and
* &lt;math&gt;\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}&lt;/math&gt; is the indicator function for the admissible set.

This equation illustrates how different constraints could theoretically be represented as restrictions on the support of a probability distribution.

It remains an abstract modeling analogy.

It does not imply that ethical rules are physical laws, that probability normalization creates temporal consistency, or that quantum mechanics requires the proposed ethical constraints.

Further work would be required to determine whether any meaningful mathematical correspondence exists between constrained probability transformations, fixed-point models, quantum channels, and hypothetical temporally consistent information systems.

Any such correspondence would need to distinguish carefully between:

* mathematical consistency within a model;
* physical realizability in nature;
* informational reliability in an engineered system; and
* ethical acceptability of an intervention.

== Conclusion ==

The central thesis is that any future information-processing system capable of interacting with causally sensitive historical information would require explicit safeguards for consistency, information integrity, and ethical intervention.

This thesis is a speculative design proposal, not a conclusion established by quantum mechanics.

Current physics provides theoretical discussions of closed timelike curves, time-symmetric descriptions, and retrocausal interpretations, but no demonstrated method for controllable communication with the past.

Current artificial intelligence and quantum-computing technologies likewise do not establish temporal interaction.

''2001: A Space Odyssey'' is therefore useful only as a fictional analogy for questions involving autonomous intelligence, information, objectives, technological capability, and system constraints.

The framework's practical relevance does not depend on time travel becoming possible.

Ordinary AI, predictive systems, augmented reality, and historical databases already create feedback between information, human behavior, and future data. Principles such as information integrity, causal minimalism, predictive transparency, and consistency checking can therefore be studied as present-day governance and design questions, while their retrocausal extension remains hypothetical.

The article consequently treats &quot;quantum ethics&quot; not as an ethical system derived from quantum theory, but as a proposed interdisciplinary vocabulary for examining how autonomous systems should behave when their actions can affect the informational conditions underlying their future decisions.

== References ==

=== Physics and temporal consistency ===

* Hawking, S. W. (1992). &quot;Chronology protection conjecture.&quot; ''Physical Review D'', 46(2), 603–611.

* Novikov, I. D. (1989). ''The River of Time''. Cambridge University Press.

=== Time-symmetric and retrocausal approaches ===

* Aharonov, Y., &amp; Vaidman, L. (2001). &quot;The Two-State Vector Formalism of Quantum Mechanics: An Updated Review.&quot; In ''Time, Quantum and Information'', or the corresponding 2001 review/preprint version. [https://arxiv.org/abs/quant-ph/0105101 arXiv:quant-ph/0105101].

=== Quantum information ===

* Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., &amp; Wootters, W. K. (1993). &quot;Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels.&quot; ''Physical Review Letters'', 70(13), 1895–1899. DOI: 10.1103/PhysRevLett.70.1895.

=== Fiction and artificial intelligence ===

* Kubrick, S. (Director). (1968). ''2001: A Space Odyssey''. Metro-Goldwyn-Mayer.

* Clarke, A. C. (1968). ''2001: A Space Odyssey''. New American Library.

== Further reading ==

The following topics provide useful background for further study:

* [[Closed timelike curve]]
* [[Novikov self-consistency principle]]
* [[Chronology protection conjecture]]
* [[Two-state vector formalism]]
* [[Quantum information]]
* [[Quantum teleportation]]
* [[Artificial intelligence]]
* [[Augmented reality]]
* [[Fixed point (mathematics)]]
* [[Dynamical system]]
* [[Constraint satisfaction problem]]
* [[Information theory]]
* [[Control theory]]

These topics provide background for the framework but should not be interpreted as evidence that the proposed temporal-ethical model is an established physical theory.

== Research status ==

This article presents a speculative interdisciplinary framework and should not be interpreted as an established physical theory.

The distinctions between established physics, theoretical interpretation, mathematical analogy, fictional analogy, and proposed ethical principles are intentional.

The mathematical formulations in this article are proposed models for further investigation. They are not presented as derivations from quantum mechanics, general relativity, or any experimentally established mechanism of time travel.

The framework is presented for discussion, criticism, collaborative development, and possible future formalization.

Future work could examine whether the proposed consistency conditions can be formalized using established mathematical tools, and whether any meaningful relationship exists between fixed-point constraints, probability transformations, information-theoretic consistency, control theory, and models of causally constrained systems.

== Author and Attribution ==

'''Original framework and article: Howard Richardson, 2026.'''

This article is made available on Wikiversity for educational discussion, collaborative development, criticism, and further research.

The original speculative framework, including the proposed connections among temporal consistency, information integrity, autonomous systems, ethical constraints, and the mathematical formulations developed in this article, is attributed to Howard Richardson.

Scientific concepts, historical material, fictional works, and published research remain attributable to their respective authors and sources.

Attribution to Howard Richardson should be retained when referring specifically to the original framework and its proposed formulations.

Wikiversity content is subject to the licensing terms displayed by the project. This attribution statement is intended to identify the origin of the original framework without asserting exclusive rights over material contributed by other authors or drawn from published sources.</text>
      <sha1>8bn2z6gm3plu31sj515q06mo6c0ucnb</sha1>
    </revision>
  </page>
  <page>
    <title>Closed timelike curve</title>
    <ns>0</ns>
    <id>332253</id>
    <revision>
      <id>2834730</id>
      <timestamp>2026-09-27T22:43:07Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>CTC subpage</comment>
      <origin>2834730</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23000" sha1="0pnfq09yyy71sb0ybsmalod03xpe2a8" xml:space="preserve">{{Research project}}

'''Closed Timelike Curves'''

'''Version 1.0 — September 2026'''

{{TOC limit|4}}

== Overview ==

A '''closed timelike curve''' (CTC) is a trajectory through spacetime that returns to an event in its own causal past. In general relativity, certain mathematical solutions of Einstein's field equations contain such curves.

The existence of these solutions is one of the reasons that general relativity permits serious theoretical discussion of unusual causal structures. It does not, however, establish that macroscopic time travel is physically realizable.

CTCs are important to the broader research framework developed in [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]] because they provide one of the clearest theoretical settings in which the concept of '''global consistency''' becomes relevant.

This page therefore separates three levels of discussion:

# mathematical and physical results concerning spacetimes containing CTCs;
# theoretical proposals for dealing with the causal paradoxes associated with CTCs; and
# the speculative extension developed in the present research project concerning information-processing systems and ethical constraints.

The third level is not presented as a consequence of general relativity.

== What Is a Closed Timelike Curve? ==

In relativity, an observer follows a timelike worldline through spacetime. A timelike curve represents a possible trajectory for an object moving locally slower than the speed of light.

A closed timelike curve is a timelike trajectory that eventually returns to the same spacetime event.

Conceptually:

&lt;blockquote&gt;
Future-directed motion → later spacetime event → continued future-directed motion → earlier spacetime event
&lt;/blockquote&gt;

The important point is that the observer does not necessarily experience their own proper time running backward. Along the timelike curve, the observer continues to move forward according to their own clock. The unusual feature is the global structure of spacetime: the worldline closes back on itself.

This creates the possibility that an observer could arrive at an event that, according to another description of the spacetime, occurs before the observer's departure.

A CTC therefore represents a possible form of '''causal self-intersection''' in spacetime.

== CTCs in General Relativity ==

General relativity describes gravity geometrically: matter and energy influence the geometry of spacetime, while that geometry determines the motion of matter and light.

The Einstein field equations admit a number of mathematical solutions with unusual causal structures.

One historically important example is the '''Gödel universe''', a rotating cosmological solution discovered by Kurt Gödel in 1949. Other theoretical examples have involved rotating spacetimes, wormhole constructions, and certain idealized configurations involving rapidly moving gravitational structures.

The existence of these mathematical solutions does not imply that the corresponding physical conditions occur in our universe.

A useful distinction is therefore:

'''Mathematical possibility.''' A spacetime geometry satisfying the relevant equations can contain CTCs.

'''Physical realizability.''' The mathematical solution must also correspond to a physically realizable configuration of matter, energy, boundary conditions, and gravitational dynamics.

'''Experimental evidence.''' There is currently no experimental evidence demonstrating a traversable macroscopic CTC.

These three statements should not be conflated.

Reviews of CTC research emphasize that general relativity contains mathematical models with chronology violation while also stressing the unresolved physical questions surrounding their formation and stability.

== Wormholes and Time-Shifted Paths ==

A frequently discussed mechanism for constructing a hypothetical time machine involves a traversable wormhole whose two mouths experience different amounts of relativistic time.

In simplified thought experiments, if one mouth is accelerated or otherwise subjected to a different spacetime history than the other, the two mouths can acquire a relative time shift.

A sufficiently large time offset could, in an idealized model, permit a trajectory entering one mouth to emerge from the other at an earlier external time.

This does not mean that traversable wormholes exist.

The construction is useful because it provides a concrete mathematical setting in which CTCs can arise and allows researchers to investigate questions concerning causality, initial conditions, quantum fields, and self-consistency.

Research on wormhole-based CTCs has also demonstrated that apparently paradoxical classical trajectories can lead to unusual questions about the existence and multiplicity of self-consistent solutions.

== The Grandfather Paradox ==

The most familiar CTC thought experiment is the '''grandfather paradox'''.

Suppose an individual travels into the past and prevents an event that is necessary for their own existence.

The resulting causal sequence appears contradictory:

&lt;blockquote&gt;
The traveler exists → the traveler travels into the past → the traveler prevents their own existence → the traveler never exists → the traveler cannot travel into the past.
&lt;/blockquote&gt;

This is an example of a '''consistency paradox'''.

Another category involves '''causal loops''', in which an object or piece of information exists in a closed causal chain without an identifiable external origin.

For example:

&lt;blockquote&gt;
A future traveler gives an invention to a person in the past → the person develops the invention → the invention is preserved → the future traveler obtains it → the traveler takes it back to the past.
&lt;/blockquote&gt;

The question then becomes:

&gt; Where did the information originally come from?

These thought experiments do not establish that CTCs exist. They illustrate why a theory permitting CTCs must address the consistency of events and information within the resulting causal structure.

== Self-Consistency and the Novikov Principle ==

One proposed response to causal paradoxes is the '''Novikov self-consistency principle'''.

In simplified form, the principle states that events occurring within a region containing a closed timelike curve must be globally self-consistent.

Under this interpretation, an observer could not use a CTC to create a contradiction in the history that contains the observer.

The grandfather paradox would therefore not be resolved by allowing both the traveler and the contradictory outcome. Instead, the globally admissible history would have to be one in which the complete sequence of events remains consistent.

This does not necessarily mean that every event is predetermined in the ordinary philosophical sense.

Rather, it imposes a constraint on which complete histories are admissible.

Research on classical systems interacting with wormhole CTCs has examined precisely this question. Echeverria, Klinkhammer, and Thorne found that apparently dangerous initial trajectories can produce multiple self-consistent solutions rather than simply having no solution.

Related work investigated the Cauchy problem in spacetimes containing CTCs and found that the presence of CTCs can alter the usual assumptions about specifying initial data and predicting future evolution.

This is an important point for the present research framework:

&gt; '''Self-consistency is a constraint on a complete causal history, not merely a rule applied to an individual event.'''

== Multiple Self-Consistent Histories ==

Self-consistency does not necessarily imply that there is only one possible history.

A mathematical model can, depending on its assumptions and initial conditions, admit more than one self-consistent trajectory.

This distinction matters because &quot;the universe must be self-consistent&quot; does not automatically mean:

&lt;blockquote&gt;
There is only one possible outcome.
&lt;/blockquote&gt;

It may instead mean:

&lt;blockquote&gt;
Only histories satisfying the relevant physical constraints are admissible.
&lt;/blockquote&gt;

This distinction becomes especially important when considering information-processing systems.

A system could, hypothetically, have several possible actions available while only some of the resulting complete histories remain consistent.

The mathematical problem would then resemble a constrained trajectory-selection problem rather than a simple deterministic prediction problem.

== Quantum Approaches to Closed Timelike Curves ==

CTCs have also been investigated using quantum mechanics.

David Deutsch proposed a quantum-mechanical treatment of systems interacting with closed timelike lines. His 1991 analysis examined how quantum mechanics might alter the paradoxes encountered in classical descriptions.

In this approach, the quantum description does not simply reproduce the classical requirement that a single classical trajectory must be followed. Instead, quantum states and consistency conditions are used to formulate the interaction.

Other research has investigated path-integral approaches and quantum effects near CTCs. Work by Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, and Yurtsever considered how quantum mechanics changes the treatment of self-consistency in systems that are problematic classically.

These approaches should not be interpreted as experimental demonstrations of quantum time travel.

They are theoretical models used to investigate what the laws of physics might imply if a spacetime containing CTCs were mathematically or physically possible.

== Chronology Protection ==

A fundamentally different response to CTCs is the possibility that nature prevents them from forming.

Stephen Hawking proposed the '''chronology protection conjecture''' in 1992.

The conjecture suggests that physical effects associated with quantum fields and gravity may prevent the formation of closed timelike curves and therefore protect causality on macroscopic scales. Hawking's original paper appeared in ''Physical Review D'' 46, 603–611.

One proposed mechanism involves the behavior of quantum fields near a '''chronology horizon''', the boundary at which closed timelike curves would begin to appear.

Calculations of vacuum polarization near such horizons have provided theoretical support for the possibility of chronology protection in some settings. Other analyses have identified circumstances in which the expected divergence may be weaker or behave differently.

Consequently, chronology protection remains a conjecture rather than an experimentally established law.

The important distinction is:

'''Self-consistency approach:'''

&lt;blockquote&gt;
CTCs may be mathematically possible, but only globally self-consistent histories can occur.
&lt;/blockquote&gt;

'''Chronology-protection approach:'''

&lt;blockquote&gt;
Physical effects may prevent CTCs from forming in the first place.
&lt;/blockquote&gt;

These approaches address the causal problem differently.

== What CTC Research Does Not Establish ==

CTC research should not be interpreted as establishing any of the following:

* that humans can travel into the past;
* that macroscopic time machines exist;
* that quantum computers can communicate with the past;
* that information can currently be transmitted backward in time;
* that wormholes exist as usable transport mechanisms;
* that extraterrestrial civilizations use CTCs; or
* that unexplained historical or contemporary observations constitute evidence for temporal intervention.

Mathematical solutions and theoretical models are valuable because they allow researchers to examine the consequences of proposed physical structures.

They do not by themselves establish that those structures occur in nature.

This distinction is central to the present research project.

== CTCs and Information ==

The most direct connection between CTC research and the broader framework of this project concerns information.

A conventional causal sequence can be represented as:

&lt;blockquote&gt;
&lt;SUB&gt;t&lt;/SUB&gt; → S&lt;SUB&gt;t+1&lt;/SUB&gt; → S&lt;SUB&gt;t+2&lt;/SUB&gt;
&lt;/blockquote&gt;

where each state depends on preceding states.

A hypothetical CTC introduces a constraint connecting a later state back to an earlier state:

&lt;blockquote&gt;
S&lt;SUB&gt;t&lt;/SUB&gt; → S&lt;SUB&gt;t+1&lt;/SUB&gt; → ... → S&lt;SUB&gt;t+n&lt;/SUB&gt; → S&lt;SUB&gt;t&lt;/SUB&gt;
&lt;/blockquote&gt;

The resulting structure is not simply a forward sequence. It is a closed trajectory.

If information is carried around the loop, then the information state must also participate in the consistency conditions.

For example:

&lt;blockquote&gt;
Information at time &lt;math&gt;t&lt;/math&gt; → future processing → later information → feedback → information at time &lt;math&gt;t&lt;/math&gt;
&lt;/blockquote&gt;

This provides a conceptual bridge from physical temporal consistency to informational consistency.

However, this bridge is a '''research proposal''', not an established physical result.

== From Physical Consistency to Informational Consistency ==

The present research framework asks whether the concept of consistency could be extended from physical trajectories to information states.

Suppose:

&lt;math&gt;S_{t+1}=F_t(S_t,A_t)&lt;/math&gt;

describes an ordinary forward transition.

A hypothetical temporally constrained system might additionally require a later state to satisfy a consistency condition involving an earlier state:

&lt;math&gt;S_t=G_t(S_{t-1},A_{t-1},S_{t+1},B_t).&lt;/math&gt;

These equations are not proposed as equations of established physics.

They are mathematical representations of a conceptual question:

&gt; '''What would it mean for an information-processing system to operate under a requirement that its complete trajectory remain self-consistent?'''

A trajectory can be represented as:

&lt;math&gt;\mathbf{S}=(S_{t_0},S_{t_0+1},\ldots,S_{t_1}).&lt;/math&gt;

A hypothetical consistency operator could then be written:

&lt;math&gt;\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).&lt;/math&gt;

The fixed-point notation expresses the idea that an admissible history reproduces itself under the complete set of transition and consistency constraints.

This is the mathematical direction developed further in the main article.

== Why This Matters for AI ==

Nothing in current CTC research establishes that artificial intelligence can interact with closed timelike curves.

The connection to AI is therefore hypothetical.

The question is instead whether an autonomous information-processing system provides a useful model for studying the consequences of global consistency constraints.

An AI system already operates through states, information, predictions, and actions.

A simplified representation is:

&lt;blockquote&gt;
Information → model state → prediction → action → changed information
&lt;/blockquote&gt;

This creates an ordinary feedback loop.

A hypothetical temporal extension would be:

&lt;blockquote&gt;
Historical information → AI model → intervention → altered historical information → future AI model
&lt;/blockquote&gt;

If physical retrocausal information transfer were ever demonstrated, such a system would raise additional questions concerning:

* information provenance;
* contradictory historical records;
* causal consistency;
* intervention boundaries;
* prediction and feedback;
* system objectives;
* ethical constraints; and
* verification of globally admissible histories.

These questions motivate the broader concept of '''temporal consistency and quantum ethics''' developed in the main research article.

They should not be interpreted as predictions that such systems will exist.

== Relationship to the Main Research Article ==

The present page provides the physical and theoretical background for the main article:

'''[[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]'''

The main article extends the discussion in a deliberately speculative direction.

Its central distinction is:

&lt;blockquote&gt;
'''CTC physics → temporal self-consistency → informational consistency → hypothetical ethical constraints'''
&lt;/blockquote&gt;

The first part of this chain belongs to established theoretical physics and the published literature on CTCs.

The later extensions are proposed conceptual models.

The framework therefore does not claim that the Novikov self-consistency principle implies an ethical theory, nor that Hawking's chronology protection conjecture implies a particular design for artificial intelligence.

Instead, the research question is:

&gt; '''If a physical system ever permitted causally sensitive information processing, could principles analogous to physical self-consistency be useful as information-governance or AI-safety constraints?'''

That question remains open.

== Research Questions ==

The CTC literature suggests several questions relevant to further development of this research project.

# '''Existence.''' Are closed timelike curves physically realizable, or are they excluded by a more complete theory of gravity?

# '''Consistency.''' If CTCs exist, what mathematical conditions determine which complete histories are admissible?

# '''Multiplicity.''' Can a given set of boundary conditions admit multiple self-consistent histories?

# '''Quantum consistency.''' How should quantum states be treated in a spacetime containing CTCs?

# '''Information.''' Can information itself be assigned consistency conditions analogous to those imposed on physical trajectories?

# '''Control.''' If a system could interact with a CTC, could its actions be constrained so that global consistency is preserved?

# '''Ethics.''' If an autonomous system could influence causally sensitive information, could ethical constraints be represented as additional admissibility conditions?

The final three questions extend beyond established CTC physics and belong to the speculative research direction of this project.

== Limitations ==

The discussion on this page has several limitations.

'''Theoretical status.''' CTCs are features of certain mathematical spacetime models. Their physical existence has not been established.

'''No demonstrated time travel.''' There is no demonstrated technology that permits travel to the past.

'''No demonstrated retrocausal communication.''' There is no established method for sending controllable information into the past.

'''Unresolved quantum-gravity question.''' General relativity and quantum field theory are not yet combined into a complete experimentally verified theory of quantum gravity capable of resolving all questions concerning chronology protection.

'''Model dependence.''' Different approaches to CTCs impose different mathematical assumptions and can lead to different conclusions.

'''Speculative AI extension.''' The connection between CTCs and autonomous artificial intelligence developed in this project is a proposed conceptual extension, not a result established by CTC research.

== Summary ==

Closed timelike curves are mathematical structures in spacetime in which a timelike trajectory can return to an earlier event.

Their existence in certain solutions of general relativity provides a legitimate theoretical setting for investigating causality and temporal consistency.

Two broad responses to the resulting causal problems are particularly important:

* the possibility that only globally self-consistent histories can occur; and
* the possibility that physical effects prevent closed timelike curves from forming.

The Novikov self-consistency principle represents the first approach, while Hawking's chronology protection conjecture represents the second.

Quantum approaches have also explored how the paradoxes and consistency conditions associated with CTCs might be represented within quantum theory. These approaches remain theoretical.

For the present research project, the most important concept is '''global consistency'''.

The proposed extension is to ask whether a similar mathematical language could eventually be used to describe information-processing systems operating under hypothetical temporal constraints.

That extension should remain clearly separated from established physics.

The purpose of this page is therefore not to establish that time travel is possible.

It is to establish the scientific foundation from which the broader speculative question can be explored:

&gt; '''If causally closed information systems were ever physically possible, what principles would be required to keep their physical states, information states, and autonomous actions globally consistent?'''

== References ==

* Deutsch, D. (1991). &quot;Quantum mechanics near closed timelike lines.&quot; ''Physical Review D'', 44(10), 3197–3217. DOI: [https://doi.org/10.1103/PhysRevD.44.3197 10.1103/PhysRevD.44.3197].

* Echeverria, F., Klinkhammer, G., &amp; Thorne, K. S. (1991). &quot;Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory.&quot; ''Physical Review D'', 44(4), 1077–1099. DOI: [https://doi.org/10.1103/PhysRevD.44.1077 10.1103/PhysRevD.44.1077].

* Friedman, J. L., Morris, M. S., Novikov, I. D., Echeverria, F., Klinkhammer, G., Thorne, K. S., &amp; Yurtsever, U. (1990). &quot;Cauchy problem in spacetimes with closed timelike curves.&quot; ''Physical Review D'', 42(6), 1915–1930. DOI: [https://doi.org/10.1103/PhysRevD.42.1915 10.1103/PhysRevD.42.1915].

* Hawking, S. W. (1992). &quot;Chronology protection conjecture.&quot; ''Physical Review D'', 46(2), 603–611. DOI: [https://doi.org/10.1103/PhysRevD.46.603 10.1103/PhysRevD.46.603].

* Kim, S.-W., &amp; Thorne, K. S. (1991). &quot;Do vacuum fluctuations prevent the creation of closed timelike curves?&quot; ''Physical Review D'', 43(12), 3929–3957. DOI: [https://doi.org/10.1103/PhysRevD.43.3929 10.1103/PhysRevD.43.3929].

* Luminet, J.-P. (2021). &quot;Closed Timelike Curves, Singularities and Causality: A Survey from Gödel to Chronological Protection.&quot; [https://arxiv.org/abs/2101.08592 arXiv:2101.08592].

== Further reading ==

* [[Closed timelike curve]]
* [[Causality]]
* [[Chronology protection conjecture]]
* [[Novikov self-consistency principle]]
* [[Wormhole]]
* [[General relativity]]
* [[Quantum mechanics]]
* [[Quantum information]]
* [[Time travel]]

== Research status ==

This page is a supporting subpage of the Wikiversity research project '''Temporal Consistency and Quantum Ethics'''.

The material concerning closed timelike curves, general relativity, quantum mechanics, Novikov self-consistency, and chronology protection is based on established published theoretical literature.

The extension from physical temporal consistency to information-processing and ethical consistency is a speculative research proposal.

The mathematical expressions used to describe that extension are conceptual formulations rather than established physical laws.

Further development should preserve the distinction between:

* established physical results;
* theoretical interpretations;
* mathematical models;
* philosophical questions; and
* original speculative proposals.

'''Original framework and research direction: Howard Richardson, 2026.'''</text>
      <sha1>0pnfq09yyy71sb0ybsmalod03xpe2a8</sha1>
    </revision>
  </page>
  <page>
    <title>Novikov self-consistency principle</title>
    <ns>0</ns>
    <id>332254</id>
    <revision>
      <id>2834731</id>
      <timestamp>2026-09-27T22:46:13Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Novikov self-consistency</comment>
      <origin>2834731</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10366" sha1="qyw3v01khzpbnpbyg3apa0bf6v2ec0n" xml:space="preserve">{{Research project}}

'''Novikov Self-Consistency Principle'''

'''Version 1.0 — September 2026'''

{{TOC limit|4}}

== Overview ==

The '''Novikov self-consistency principle''' is a proposed principle for dealing with causal paradoxes in physical models containing [[closed timelike curves]] (CTCs).

In its basic form, the principle states that if a physical process occurs within a spacetime containing a closed timelike curve, the events along that curve must be globally self-consistent.

A local physical solution cannot simply be combined with other events in a way that produces a contradiction. Instead, an admissible local solution must be extendable into a complete global solution satisfying the relevant physical laws throughout the spacetime.

This principle is associated with physicist [[Igor Novikov]] and with subsequent work by Novikov and other researchers on the Cauchy problem in spacetimes containing closed timelike curves.

The principle does '''not''' establish that closed timelike curves exist.

It does not establish that time travel is technologically possible.

It is instead a proposed consistency condition for theories or models in which closed timelike curves are already assumed to be possible.

== Historical and Theoretical Background ==

The motivation for the self-consistency principle arises from the causal paradoxes associated with closed timelike curves.

If a worldline could return to its own causal past, it might appear possible to perform an action that prevents the conditions necessary for that action to occur.

The familiar example is the grandfather paradox:

&lt;blockquote&gt;
A traveler goes into the past and prevents an event necessary for the traveler's own existence.
&lt;/blockquote&gt;

The resulting sequence appears contradictory:

&lt;blockquote&gt;
Traveler exists → traveler travels into the past → traveler prevents own existence → traveler does not exist → traveler cannot have traveled into the past.
&lt;/blockquote&gt;

The Novikov approach does not solve this paradox by allowing contradictory histories.

Instead, it imposes a consistency requirement on the complete history.

Only histories in which all events are mutually compatible would be physically admissible under the principle.

== The Formal Idea of Global Self-Consistency ==

The important word in the principle is '''global'''.

A local solution may appear physically reasonable when examined over a small region of spacetime.

However, if that local solution cannot be extended into a complete solution satisfying the physical equations throughout the relevant spacetime, it would not represent an admissible history.

The 1990 paper by Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, and Yurtsever formulated this idea explicitly. They proposed that, if closed timelike curves are permitted, the laws of physics should be supplemented by a self-consistency principle requiring a local solution to be extendable into a global solution that is well-defined throughout the nonsingular regions of spacetime.

This can be represented conceptually as:

&lt;blockquote&gt;
Local solution → global extension → consistency check → admissible history
&lt;/blockquote&gt;

The principle therefore changes the usual intuition about initial conditions.

In an ordinary causal spacetime, sufficiently specified initial data can be evolved forward according to the equations of motion.

In a spacetime containing CTCs, the complete solution may impose additional restrictions because later portions of the trajectory can be connected causally to earlier portions.

== Self-Consistent Histories ==

A self-consistent history is a complete physical history in which the events occurring along the closed causal structure do not contradict one another.

Suppose a hypothetical system has a state trajectory:

&lt;math&gt;\mathbf{S}=(S_{t_0},S_{t_0+1},\ldots,S_{t_1}).&lt;/math&gt;

In an ordinary forward-evolution model:

&lt;math&gt;S_{t+1}=F_t(S_t,A_t).&lt;/math&gt;

A closed causal structure could additionally impose a boundary condition connecting the final portion of the trajectory back to an earlier portion.

Conceptually:

&lt;math&gt;S_{t_0}=G(S_{t_1}).&lt;/math&gt;

The precise mathematical form of such a condition would depend on the physical model.

The important conceptual point is that the beginning and end of the trajectory can no longer be treated as completely independent.

The complete trajectory must satisfy the entire set of constraints simultaneously.

A compact mathematical analogy is therefore:

&lt;math&gt;\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).&lt;/math&gt;

Here &lt;math&gt;\Phi&lt;/math&gt; represents the complete set of transition and consistency conditions.

A self-consistent history is represented by a fixed point of that operation.

This fixed-point notation is a mathematical analogy used in the present research project. It should not be interpreted as the original formulation of Novikov's principle or as an established equation of physics.

== The Principle Does Not Necessarily Imply a Single History ==

An important point is that self-consistency does not automatically mean that there is exactly one possible history.

Classical models involving closed timelike curves can admit more than one self-consistent solution for a given set of initial conditions.

Echeverria, Klinkhammer, and Thorne investigated a billiard-ball model involving a wormhole that produces a closed timelike curve. Their analysis found cases in which initial trajectories could have more than one self-consistent continuation.

This means that the principle is better understood as a restriction on '''admissibility''' than as a claim of unique determinism.

Conceptually:

&lt;blockquote&gt;
Possible histories → consistency constraints → admissible histories
&lt;/blockquote&gt;

There may be one admissible history, several admissible histories, or, depending on the model and assumptions, difficulties in finding any admissible solution for particular initial conditions.

The existence and multiplicity of solutions are therefore separate mathematical questions.

== The Billiard-Ball Example ==

The billiard-ball model developed by Echeverria, Klinkhammer, and Thorne provides a useful illustration.

Imagine a billiard ball entering a wormhole and emerging at an earlier time.

The ball could potentially encounter its earlier self.

One might imagine that the later ball could collide with the earlier ball in such a way that the earlier ball never enters the wormhole.

That appears paradoxical.

However, a self-consistent solution can instead involve the collision changing the earlier ball's trajectory in precisely the way necessary for the ball to enter the wormhole and eventually produce the collision.

The resulting trajectory can therefore be globally consistent even though it involves self-interaction.

The important lesson is not that the universe &quot;forces&quot; the ball to behave intelligently.

Rather, the equations and boundary conditions admit only trajectories satisfying the complete consistency requirements.

Echeverria, Klinkhammer, and Thorne found multiple self-consistent classical solutions for some initial trajectories in their model.

This illustrates why the phrase '''global self-consistency''' is more precise than simply saying that &quot;the past cannot be changed.&quot;

The past can be part of a causal loop in the model, but the complete loop must remain mutually consistent.

== Novikov's Later Formulation ==

Novikov's 1992 paper, &quot;Time machine and self-consistent evolution in problems with self-interaction,&quot; examined physical processes involving self-interactions in spacetimes containing closed timelike curves and presented examples of self-consistent solutions.

The work develops the idea that a time-machine spacetime does not automatically lead to logical contradiction.

Instead, the equations of motion and the global structure of spacetime can restrict the allowed solutions.

Later work by Novikov and collaborators investigated whether the principle of self-consistency could arise from action principles in certain models. These studies considered self-interacting particles in wormhole spacetimes and found that stationary-action trajectories could coincide with globally self-consistent trajectories under particular assumptions.

These results should be interpreted within the mathematical models in which they were obtained.

They do not establish that nature contains usable time machines.

== Self-Consistency Is Not the Same as Determinism ==

It is useful to distinguish '''self-consistency''' from '''determinism'''.

A deterministic theory can, under suitable conditions, assign a unique future state to a specified initial state.

A self-consistency condition instead asks whether a complete trajectory satisfies all of the constraints imposed by the physical model.

In a CTC setting, these concepts can come apart.

For example, a particular set of initial data might admit:

* one self-consistent continuation;
* multiple self-consistent continuations; or
* under some classical models, no self-consistent continuation.

Consequently, the statement

&gt; &quot;The history must be self-consistent&quot;

does not necessarily mean

&gt; &quot;There is only one possible history.&quot;

This distinction becomes important when considering whether a hypothetical autonomous system could &quot;choose&quot; among possible histories.

The physics of the underlying model would have to determine what choices, if any, are meaningful.

== Self-Consistency and Information ==

The original principle concerns physical solutions and causal histories.

The present research project asks whether the same mathematical idea might provide a useful analogy for '''information consistency'''.

Consider an ordinary information-processing system:

&lt;blockquote&gt;
State → information processing → action → new state
&lt;/blockquote&gt;

Now suppose the system's later state could somehow influence an earlier informational state:

&lt;blockquote&gt;
Earlier information → system processing → later information → feedback → earlier information
&lt;/blockquote&gt;

A hypothetical consistency condition might require the complete informational trajectory to remain mutually compatible.

This is not part of the established Novikov principle.

It is a proposed extension of its conceptual structure.

The distinction is therefore:

'''Novikov self-consistency:'''

&gt; A proposed physical</text>
      <sha1>qyw3v01khzpbnpbyg3apa0bf6v2ec0n</sha1>
    </revision>
  </page>
  <page>
    <title>Plant Divisions (Phyla)/Glaucophyta</title>
    <ns>0</ns>
    <id>332255</id>
    <revision>
      <id>2834738</id>
      <timestamp>2026-09-27T23:44:26Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Created page with &quot;[[File:Glaucophyte.jpg|thumb|300x300px|This is a Glaucophyte.]] The '''glaucophytes''', aka '''glaucocystophytes''' or '''glaucocystids''', are a small group of unicellular [[Streptophytes]] found in freshwater and moist terrestrial environments,&lt;ref name=&quot;keeling&quot;&gt;{{cite journal |journal =[[American Journal of Botany]] |year=2004 |volume=91 | pages=1481–1493 | title=Diversity and evolutionary history of plastids and their hosts |last=Keeling |first=Patrick J. | doi=10...&quot;</comment>
      <origin>2834738</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2598" sha1="lyw6v5iz4o9nh9pcpqd6tpkz4twimzj" xml:space="preserve">[[File:Glaucophyte.jpg|thumb|300x300px|This is a Glaucophyte.]]
The '''glaucophytes''', aka '''glaucocystophytes''' or '''glaucocystids''', are a small group of unicellular [[Streptophytes]] found in freshwater and moist terrestrial environments,&lt;ref name=&quot;keeling&quot;&gt;{{cite journal |journal =[[American Journal of Botany]] |year=2004 |volume=91 | pages=1481–1493 | title=Diversity and evolutionary history of plastids and their hosts |last=Keeling |first=Patrick J. | doi=10.3732/ajb.91.10.1481 |issue=10 |pmid=21652304 |doi-access=free |bibcode=2004AmJB...91.1481K }}&lt;/ref&gt;&lt;ref&gt;[https://books.google.com/books?id=lE6r5q5op94C&amp;dq=%22Glaucophyta+%28also+known+as+Glaucocystophyta%29+is+a+small+and+inconspicuous+group+of+unicellular+algae+found+in+freshwater+and+terrestrial+environments%22&amp;pg=PA74 Genomic Insights Into the Biology of Algae]&lt;/ref&gt;but they are less common today than they were during the [[Wikipedia:Proterozoic|Proterozoic]].&lt;ref name=&quot;Cruzan 2018&quot;&gt;{{cite book |last=Cruzan |first=Mitchell B. |title=Evolutionary Biology |publisher=Oxford University
 Press |year=2018 |isbn=978-0-19-088268-6 |page=20 }}&lt;/ref&gt;
The glaucophytes are of interest to biologists studying the evolution of chloroplasts as they may be similar to the ancestral algal type that led to the red algae and green plants.
==Information==
Name Meaning: Blue-green plant

English Common Name: Glaucophytes

Major distinguishing characteristics:

Approximate number of species described: 14 - 26&lt;ref&gt;[http://jcs.biologists.org/content/131/2/jcs203414 The monoplastidic bottleneck in algae and plant evolution | Journal of Cell Science]&lt;/ref&gt;
==Phylogeny==
Here's the phylogeny. The relationship among the three groups remains uncertain, although it is most likely that glaucophytes diverged first:

{{clade
|label1=Plants
|1={{clade
   |1='''Glaucophytes'''
[[File:Glaucocystis sp.jpg|60px]]
   |2={{clade
      |1=Red algae
[[File:Red algae 3.jpg|60px]]
      |2=Green Plants
[[File:Tionesta-ac-moss2.jpg|60px]]
      }}
   }}
}}
{{Clade
|label1='''Glaucophytes'''
|1={{Clade
  |label1=Cyanophoraceae
  |1=''Cyanophora''
[[File:Woelfib cyanphoraparadoxa teilungsfigur 1 0632002 img 54414492 ude 20131024233254 small.jpg|60px]]
  |2={{Clade
    |label1=Gloeochaetaceae
    |1={{Clade
      |1=''Cyanoptyche''
      |2=''Gloeochaete''[[File:Gloeochaete wittrockiana 336295979.jpg|60px]]
       }}
    |label2=Glaucocystidaceae
    |2={{Clade
      |1=''Glaucocystopsis''
      |2=''Glaucocystis''
[[File:41598 2015 Article BFsrep14735 Fig1a-Glaucocystis geitleri.jpg|60px]]
       }}
     }}
   }}
}}
==References==</text>
      <sha1>lyw6v5iz4o9nh9pcpqd6tpkz4twimzj</sha1>
    </revision>
    <revision>
      <id>2834739</id>
      <parentid>2834738</parentid>
      <timestamp>2026-09-27T23:45:21Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <origin>2834739</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2681" sha1="gmr7bu8kmit4qelfc8lsltl3iw7xg8z" xml:space="preserve">[[File:Glaucophyte.jpg|thumb|300x300px|This is a Glaucophyte.]]
The '''glaucophytes''', aka '''glaucocystophytes''' or '''glaucocystids''', are a small group of unicellular [[Streptophytes]] found in freshwater and moist terrestrial environments,&lt;ref name=&quot;keeling&quot;&gt;{{cite journal |journal =[[American Journal of Botany]] |year=2004 |volume=91 | pages=1481–1493 | title=Diversity and evolutionary history of plastids and their hosts |last=Keeling |first=Patrick J. | doi=10.3732/ajb.91.10.1481 |issue=10 |pmid=21652304 |doi-access=free |bibcode=2004AmJB...91.1481K }}&lt;/ref&gt;&lt;ref&gt;[https://books.google.com/books?id=lE6r5q5op94C&amp;dq=%22Glaucophyta+%28also+known+as+Glaucocystophyta%29+is+a+small+and+inconspicuous+group+of+unicellular+algae+found+in+freshwater+and+terrestrial+environments%22&amp;pg=PA74 Genomic Insights Into the Biology of Algae]&lt;/ref&gt;but they are less common today than they were during the [[Wikipedia:Proterozoic|Proterozoic]].&lt;ref name=&quot;Cruzan 2018&quot;&gt;{{cite book |last=Cruzan |first=Mitchell B. |title=Evolutionary Biology |publisher=Oxford University
 Press |year=2018 |isbn=978-0-19-088268-6 |page=20 }}&lt;/ref&gt;
The glaucophytes are of interest to biologists studying the evolution of chloroplasts as they may be similar to the ancestral algal type that led to the red algae and green plants.
==Information==
Name Meaning: Blue-green plant

English Common Name: Glaucophytes

Major distinguishing characteristics:

Approximate number of species described: 14 - 26&lt;ref&gt;[http://jcs.biologists.org/content/131/2/jcs203414 The monoplastidic bottleneck in algae and plant evolution | Journal of Cell Science]&lt;/ref&gt;
==Phylogeny==
Here's the phylogeny. The relationship among the three groups remains uncertain, although it is most likely that glaucophytes diverged first:

{{clade
|label1=Plants
|1={{clade
   |1='''Glaucophytes'''
[[File:Glaucocystis sp.jpg|60px]]
   |2={{clade
      |1=Red algae
[[File:Red algae 3.jpg|60px]]
      |2=Green Plants
[[File:Tionesta-ac-moss2.jpg|60px]]
      }}
   }}
}}
{{Clade
|label1='''Glaucophytes'''
|1={{Clade
  |label1=Cyanophoraceae
  |1=''Cyanophora''
[[File:Woelfib cyanphoraparadoxa teilungsfigur 1 0632002 img 54414492 ude 20131024233254 small.jpg|60px]]
  |2={{Clade
    |label1=Gloeochaetaceae
    |1={{Clade
      |1=''Cyanoptyche''
      |2=''Gloeochaete''[[File:Gloeochaete wittrockiana 336295979.jpg|60px]]
       }}
    |label2=Glaucocystidaceae
    |2={{Clade
      |1=''Glaucocystopsis''
      |2=''Glaucocystis''
[[File:41598 2015 Article BFsrep14735 Fig1a-Glaucocystis geitleri.jpg|60px]]
       }}
     }}
   }}
}}
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>gmr7bu8kmit4qelfc8lsltl3iw7xg8z</sha1>
    </revision>
    <revision>
      <id>2834752</id>
      <parentid>2834739</parentid>
      <timestamp>2026-09-28T00:54:29Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <origin>2834752</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2726" sha1="c1oh890emqkccot8twt8ghl7aoyxqyo" xml:space="preserve">[[File:Glaucophyte.jpg|thumb|300x300px|This is a Glaucophyte.]]
The '''glaucophytes''', aka '''glaucocystophytes''' or '''glaucocystids''', are a small group of unicellular [[Streptophytes]] found in freshwater and moist terrestrial environments,&lt;ref name=&quot;keeling&quot;&gt;{{cite journal |journal =[[American Journal of Botany]] |year=2004 |volume=91 | pages=1481–1493 | title=Diversity and evolutionary history of plastids and their hosts |last=Keeling |first=Patrick J. | doi=10.3732/ajb.91.10.1481 |issue=10 |pmid=21652304 |doi-access=free |bibcode=2004AmJB...91.1481K }}&lt;/ref&gt;&lt;ref&gt;[https://books.google.com/books?id=lE6r5q5op94C&amp;dq=%22Glaucophyta+%28also+known+as+Glaucocystophyta%29+is+a+small+and+inconspicuous+group+of+unicellular+algae+found+in+freshwater+and+terrestrial+environments%22&amp;pg=PA74 Genomic Insights Into the Biology of Algae]&lt;/ref&gt;but they are less common today than they were during the [[Wikipedia:Proterozoic|Proterozoic]].&lt;ref name=&quot;Cruzan 2018&quot;&gt;{{cite book |last=Cruzan |first=Mitchell B. |title=Evolutionary Biology |publisher=Oxford University
 Press |year=2018 |isbn=978-0-19-088268-6 |page=20 }}&lt;/ref&gt;
The glaucophytes are of interest to biologists studying the evolution of chloroplasts as they may be similar to the ancestral algal type that led to the red algae and green plants.
==Information==
Name Meaning: Blue-green plant

English Common Name: Glaucophytes

Major distinguishing characteristics:

Approximate number of species described: 14 - 26&lt;ref&gt;[http://jcs.biologists.org/content/131/2/jcs203414 The monoplastidic bottleneck in algae and plant evolution | Journal of Cell Science]&lt;/ref&gt;
==Orders==
Glaucocystales – Gloeochaetales
==Phylogeny==
Here's the phylogeny. The relationship among the three groups remains uncertain, although it is most likely that glaucophytes diverged first:

{{clade
|label1=Plants
|1={{clade
   |1='''Glaucophytes'''
[[File:Glaucocystis sp.jpg|60px]]
   |2={{clade
      |1=Red algae
[[File:Red algae 3.jpg|60px]]
      |2=Green Plants
[[File:Tionesta-ac-moss2.jpg|60px]]
      }}
   }}
}}
{{Clade
|label1='''Glaucophytes'''
|1={{Clade
  |label1=Cyanophoraceae
  |1=''Cyanophora''
[[File:Woelfib cyanphoraparadoxa teilungsfigur 1 0632002 img 54414492 ude 20131024233254 small.jpg|60px]]
  |2={{Clade
    |label1=Gloeochaetaceae
    |1={{Clade
      |1=''Cyanoptyche''
      |2=''Gloeochaete''[[File:Gloeochaete wittrockiana 336295979.jpg|60px]]
       }}
    |label2=Glaucocystidaceae
    |2={{Clade
      |1=''Glaucocystopsis''
      |2=''Glaucocystis''
[[File:41598 2015 Article BFsrep14735 Fig1a-Glaucocystis geitleri.jpg|60px]]
       }}
     }}
   }}
}}
==References==
[[Category:Plants]]
[[Category:Taxonomy]]
[[Category:Botany]]
[[Category:Biology]]</text>
      <sha1>c1oh890emqkccot8twt8ghl7aoyxqyo</sha1>
    </revision>
  </page>
  <page>
    <title>Plant Divisions (Phyla)/Gnetophyta</title>
    <ns>0</ns>
    <id>332256</id>
    <revision>
      <id>2834754</id>
      <timestamp>2026-09-28T01:10:08Z</timestamp>
      <contributor>
        <username>The Citer</username>
        <id>3110681</id>
      </contributor>
      <comment>Ok...</comment>
      <origin>2834754</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2267" sha1="evwqzypawnud62lihlqivm3krohyscv" xml:space="preserve">[[File:Welwitschia at Ugab River basin.jpg|thumb|300x300px|This is a Gnetophyte]]
'''Gnetidae''', cka '''gnetophytes''', is a subclass of gymnosperms within the class Pinopsida.&lt;ref&gt;{{Cite web |title=''Gnetidae'' Pax |url=https://www.worldfloraonline.org/taxon/wfo-4100002555 |website=[[World Flora Online]]}}&lt;/ref&gt; The group was previously considered the distinct class '''Gnetopsida''' within the distinct division '''Gnetophyta'''. However, nuclear phylogenomic studies recover gnetophytes as the sister group of the Pinaceae, which means that gnetophytes are nested within Pinopsida, meaning they are technically [[Plant Divisions (Phyla)/Pinophyta (Coniferophyta)|pinophytes]].{{How}}
==Information==
Name Meaning: Gnetum-like plant

English Common Name: Gnetophytes

Major distinguishing characteristics: Seeds and woody vascular system with vessels.

Approximate number of species described: 70
==Evolutionary History==
Though some fossils that have been proposed to be gnetophytes have been found as far back as the Permian,&lt;ref name=&quot;Zi-Qiang Wang-2004&quot;&gt;{{cite journal |author=Zi-Qiang Wang |year=2004 |title=A New Permian Gnetalean Cone as Fossil Evidence for Supporting Current Molecular Phylogeny |url= |journal=Annals of Botany |volume=94 |issue=2 |pages=281–288 |doi=10.1093/aob/mch138 |pmc=4242163 |pmid=15229124}}&lt;/ref&gt; their affinities to the group are [[wiktionary:equivocal|equivocal]]. The oldest fossils that are certainly assignable to the group date to the Late Jurassic.&lt;ref name=&quot;Coiro-2022&quot;&gt;{{Cite journal |last1=Coiro |first1=Mario |last2=Roberts |first2=Emily A. |last3=Hofmann |first3=Christa-Ch. |last4=Seyfullah |first4=Leyla J. |date=2022-12-14 |title=Cutting the long branches: Consilience as a path to unearth the evolutionary history of Gnetales |journal=Frontiers in Ecology and Evolution |volume=10 |article-number=1082639 |doi=10.3389/fevo.2022.1082639 |issn=2296-701X |doi-access=free |bibcode=2022FrEEv..1082639C }}&lt;/ref&gt; Overall, the fossil record of the group is richest during the Early Cretaceous, exhibiting a substantial decline during the Late Cretaceous.&lt;ref name=&quot;Coiro-2022&quot; /&gt;
==Orders==
Since Gnetophyta is not a divison anymore, let's see its orders:&lt;br&gt;
Ephedrales – Gnetales – Welwitschiales
==References==</text>
      <sha1>evwqzypawnud62lihlqivm3krohyscv</sha1>
    </revision>
  </page>
  <page>
    <title>Special pages</title>
    <ns>0</ns>
    <id>332257</id>
    <redirect title="Special:SpecialPages" />
    <revision>
      <id>2834793</id>
      <timestamp>2026-09-28T04:43:08Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>#REDIRECT [[Special:SpecialPages]]</comment>
      <origin>2834793</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="34" sha1="29k3tj7v55mrtgpfiay9zpkpj1ndcwt" xml:space="preserve">#REDIRECT [[Special:SpecialPages]]</text>
      <sha1>29k3tj7v55mrtgpfiay9zpkpj1ndcwt</sha1>
    </revision>
  </page>
</mediawiki>